The question presented is "on what dimensions/issues does the Republican party differs substantially from right-wing parties in Europe?"
An answer to this question has to start from the point of recognizing a false premise in the question. The U.S. has one viable party of the left, the Democratic Party, and one viable party of the right, the Republican Party. Europe, with minor exceptions, has two of each. Most countries in Europe, in contrast, have a far left party, a center-left party, a center-right party, and a far right party, in addition to some additional parties (see also, e.g. here and here and here).
So, to properly answer this question one really has to address two points: how does the Republican Party differ from European Center-Right Parties and how does the Republican Party differ from European Far Right Parties.
The short answer is that after the Republican and Democratic parties traded positions on the left-right political scale (roughly from the 1960s to the 1980s), the Republican party was originally closer to a Center-Right party of Europe. But the U.S. Republican Party has moved further to right over time after the major political parties in the U.S. traded ideological positions, and it is now closer right now to a European far right party than to a European center-right party right now.
No analogy between the U.S. Republican Party and European parties of the right is completely apt, however. The U.S. Republican party it has a different relationship to the business establishment and government institutions than either European center-right parties or European far right parties.
Republicans have similar attitudes towards ethic diversity, political violence, distrust of the media, and authoritarianism to the far right parties of Europe. See, e.g. here.
The U.S. Republican Party is as or more disdainful of existing government institutions, even usually historically uncontroversial ones like the Post Office, than any party anywhere on the political spectrum in Europe and is closer to the far right than the center-right of Europe in this regard. See, e.g. here (comparing the Tea Party faction of the GOP to the far right parties of Europe).
The U.S. Republican Party used to have a relationship with the business establishment similar to that of most center-right parties in Europe. And, the U.S. Republican Party still generously favors many business interests in deregulation and low taxes. But unlike European Center-Right parties the Republican Party currently lacks almost any inclination to invest in infrastructure or adopt affirmative regulatory regimes conducive to business interests. Also, the existing U.S. Republican party, which previously had a pro-Big Business globalist outlook, now shares the deep anti-globalism and anti-immigration stances of the European far right. See, e.g., here.
The U.S. Republican Party also have a strong vein of anti-intellectualism, distrust of science, and distaste for academia and experts that has no rival anywhere on the spectrum of partisan elected officials in Europe.
For example, many European Center-Right parties are skeptical of government education spending because they see it is inefficient and wasteful. The U.S. Republican Party in contrast, is more concerned that the worldview of academia is not aligned with its own anti-intellectual, Evangelical Christian centered worldview.
The place to begin the comparison is the recognize that in most of Europe there are multiple viable political parties, rather than just two as there are in the United States. This complicates comparisons between U.S. political parties and European ones and most be explained at some length for any comparison to provide much understanding.
A brief overview of the history of European political parties
At the dawn of World War II (i.e. 1939) in Europe, the feudal systems and absolute monarchies that have been predominant outside a few small city-states and small outlier countries since the Middle Ages had been predominantly replaced by constitutional monarchies with elected legislatures of some type, democratically elected multiparty republics, and in parts of Eastern Europe, one party Soviet Style Communist republics. The constitutional monarchy in England and the First Republic of France (1789) were the first significant examples, but absolute monarchies were still common. Some early attempts at Republican government (such as France) failed the first time around and uprisings against monarchies in many countries across Europe in 1848 failed. But the tide turned in the late 19th century into the very early 20th century. Many countries followed a model imposed upon them by Napoleon, but there was also great local variation. The modern democratic systems (and national boundaries) of German and Italy, for example, date to the 1870s.
Outside of the United Kingdom, Ireland, Sweden, Finland, Switzerland, Portugal and Spain, the ordinary democratic government process in Europe was largely interrupted during World War II.
After World War II, most countries in Europe whose democratic processes were interrupted either adopted new constitutions from scratch, or reinstated and "rebooted" old constitutions (often with significant constitutional amendments) refilling all elected offices from scratch. The new constitutions and amendments to old ones, where adopted were based upon prevailing ideas about the best forms of government at the time, which were widely shared across Western Europe, such as the virtues of proportional representation and constitutional protections of human rights (often in the form of a special constitutional court or council).
The first round of post-WWII elections in each country necessarily involved the formation, or "re-booting" of political parties in each country, greatly reshuffling the organization and coalitions backing each party. To generalize (as one must in a short answer like this one), most countries had a couple of left leaning parties, a couple of right leaning parties and usually a small number of additional parties that survived the initial weeding out and consolidation of new political parties in the Post-WWII era, and there have been select instances of new party formation and old party death since then, especially in the post-Cold War era which had led to new progressive parties (sometimes replacing or marginalizing former Marxist-Leninist parties), new "far right" parties, and occasionally new moderate parties (e.g. in Greece when an existing major was discredited).
Eastern Europe mostly transitioned into one party, Soviet style Communist systems on the model of the Soviet Union, until the Cold War ended, and then these countries adopted Western European style constitutions giving rise to a wave of political party formation analogous to that of Continental Western Europe after WWII, but informed by the previous half century of Western European political party formation. The wave of new political party formation in Eastern Europe may have been one factor leading to a much smaller wave of political party formation and innovation in Western Europe at around the same time.
The main generic types of political parties in Europe
In most of these countries, the multiparty political systems that emerged (as opposed to a one party system, dominant party system, or two dominant party system) that emerged was largely a function of some manner of proportional representation electoral system in the context of a parliamentary system on the Westminster model of the United Kingdom, in which governing majority coalitions are formed after the election by members of parliament, rather than before the election, and in which parties have much more control of their legislative agenda and the candidates that will represent them than the do in the United States.
Historically, the two largest parties in European political systems are a center-left party, known in England as the Labour Party and in most of the rest of Europe as a Democratic Socialist Party, and a center-right party, known in England as the Conservative Party (a.k.a. the Tories) and in the rest of Europe, most often as a Christian Democratic Party (the center-right parties are Europe join together in a party known as the EPP at the EU level).
Historically (i.e. roughly speaking from after WWII until the years immediately following the end of the Cold War), casual observers would identity the Democratic Party with the Center-Left parties of Europe and would identify the Republican Party with the Center-Right parties of Europe.
Usually, there is a "far right" party23 to the right of the Center Right Party whose names vary considerably but which tend to be opposed to globalism, to be xenophobic, to be pro-authoritarian and to have a degree of distrust of the socio-economic, globalist, financial sector and big business dominated establishment. (See also here and here.)
Usually, there is a "far left" party to the left of the Center-Left Party often called the "Green Party" which corresponds more or less to the factions in the U.S. Democratic party and in left leaning third-parties and in movement politics outside of political parties in the U.S. often described these days as "Progressive." See also here. Some European countries have an old style Marxist-Leninist Communist Party to the left of the Green Parties.
Frequently, there is a political party between the center-left party and the center-right party on a left to right political divide. In England and Wales, this is called the Liberal Party which seeks to be socially liberal while economically free market oriented. Sometimes, instead, there will be a party sometimes called a "Center Party" (also here) or a party built around an iconoclastic political figure not easily pigeon holed into left or right wing labels instead.
Finally, in many countries there are often regional nationalist parties in areas such as Scotland, Northern Ireland and Basque Spain whose substantive policy views may very but which seek greater regional autonomy or independence for an ethnically and geographically distinct geographical region within a larger country.
Thumbnail sketches of the political parties of Europe at the E.U. level can be found here.
The United Kingdom's single member plurality district electoral system, that lacks either proportional representation or ranked choice voting or runoff elections if no one candidate attains a majority, causes its far left and far right political parties to be particularly weak, causes the centrists party to typically receive fewer seats in parliament than its share of the popular vote, encourages regional nationalist political parties, and strengthens its center-left and center-right parties relative to the European norm.
The implications of a multiparty system
The multiparty system common in Europe has multiple consequences.
Political Parties typically have better defined and more coherent agendas, because no one party has to please a majority of voters.
Divided government is rare (except in France which has a fairly strong Presidency and a particularly complex constitutional order).
The most powerful position for an elected official is the Prime Minister who is typically the leader of the largest of the political parties (by number of legislators in parliament) in a ruling coalition of more than one political party. This coalition is generally assembled after an election, reaches a formal internal charter on policy positions on key issues upon which the coalition member parties may differ in platform, and allocates government ministries (the equivalent of U.S. government departments led by a Secretary, for example, of Agriculture) more or less proportionately to the number of legislators of each coalition member's party allocated roughly based upon which party has the most intense interest in each post that is not at odds strongly with other coalition member parties. The ministers in each party then have fairly broad discretion to set policy in their allocated ministries consistent with the coalition charter.
Since Prime Ministers are chosen indirectly by their fellow members of parliament, the selection usually represents a much more informed evaluation of the personal character of that individual and parliament can withdraw support from a prime minister who is screwing up without an election, so the Prime Minister has less freedom of action politically than a popularly elected President responsible only to the electorate at the next election.
Since the party has more control over who runs for office under its banner and who will serve in cabinet (or as a "shadow cabinet" member) as opposed to being a "back bencher" the party has more power to implement the political platform adopted by rank and file party members than a U.S. political party.
It is easier for a fairly modest sized but ideologically coherent group of people in Europe who are a minority of the overall electorate to get a seat at the political table, a prominent political voice, and potentially a junior position of a governing coalition that allows that party to have a role in setting policy in some subject matter area, than in the U.S. system where every politician must appeal to a majority in a geographic area. In Europe, diffuse but coherent political agendas are represented in parliament, in the U.S. someone who holds a minority political view from a national perspective needs to win a majority in some reasonable compact geographic district (relative to the total size of the country or state).
In Europe's electoral system favorable to a multiparty political landscape, the natural way to resolve a deep divide over policy or political tactics is to have each faction form a political party and then to form a coalition if the other faction is still your most likeminded ally.
In contrast, in the U.S. political party system, the way that an ideological faction within a political party gains power is to strategize to win key political party and elective offices within one of the two major parties, especially the top of the ticket Presidency.
Also, for a variety of reasons, it is less expensive to run a viable political campaign in Europe than in the U.S. and parties play a more central role in fundraising collectively for all of its members. In contrast, in the U.S. system (by design) candidates must personally raise almost all campaign funds and it is expensive to run a campaign in a competitive district.
The U.S. Republican Party shifted from being a centerist or center-left party to a center right party in the 1960s to 1980s, and then shifted after that time period further right to be (mostly) a far right party
The realignment of the U.S. Republican and Democratic parties
The Republican party was formed in the years leading up to the U.S. Civil War as an anti-slavery center-left party. During the years of post-Civil War reconstruction and for many decades afterwards, the Republican party was the more liberal party in U.S. politics, and the Democratic party was the more conservative party and the dominant party of Southern whites. But in the wake of "realignment" starting in the 1960s or so, which had mostly run its course by roughly the late 1980s but continuing to the present, the Democratic Party became the party of the left and the Republican Party became the party of the right and the dominant party among white Southerners, trading the ideological spaces that they had held a century earlier.
The post-realignment evolution of the Democratic and Republican parties in the U.S.
In the U.S. two party system, the party of the left and the party of the right, each strains to sufficiently satisfy everyone near majority coalitions formed prior to the election, so they can win control in elections. Each of the major two U.S. national political parties needs three things to function and survive.
A coalition that is sufficiently broad to have a credible chance of controlling both houses of Congress and the Presidency in each biannual federal election. Both policy and culturally driven political identity drive the content of these coalitions as do issues of which factions are compatible with each other.
One or more groups of people in the coalition capable of providing meaningful funding for political campaigns and party operations. Post-realignment, Democrats have relied upon liberal professionals like lawyers for left leaning interests and affluent non-Christians for their cash, and secondarily on unions, while Republicans have relied upon big businesses and affluent individuals and had more money. Now, the two major parties share big business sources of finance and Internet based small progressive donations provide much of the Democrats' financial support.
One or more groups of people in the coalition who can provide a mass of grass roots volunteers. Post-realignment, Democrats have relied upon unions for this and Republicans have relied upon the Christian Right. As union ranks have gotten smaller, Democrats have turned to grass roots progressive activists and identity-politics motivated volunteers.
The Democratic Party has evolved into a center-left party that also has a progressive party faction within it. Meanwhile, the Republican Party started out after realignment as a center-right party has overtime increasingly become a far right and White Christian Nationalist party, at the expense of some of its establishment wing.
The migration of some economic establishment figures to the Democratic Party, and of white working class voters to the Republican Party with which they identify culturally, as the blue collar union movement has weakened, has pulled the Democratic Party in a moderate direction economically, and shifted the Republican Party's historic role as the party of the economic establishment to become a culturally united party of the right.
The fight for the soul of the Republican party
The Republican Party has responded to conflict between its factions by basically conducting a duel between the center-right economic establishment and moderate conservative mainline Christians and Roman Catholics on one side, and far right cultural conservatives centered around working class Evangelical Christians from the South and rural American on the other, while quietly throwing some tax breaks and deregulation measures at its economic establishment wing now and then.
The far right started the war for the heart of the Republican Party with anti-abortion activism but escalated it to an all out far right cultural agenda in terms of self-identity and rhetoric with the Trump Presidency.
The Republican Party in the U.S. is far less committed emotionally and in priorities to the globalist oriented economic establishment than it used to be and far less than most center-right parties of Europe that by and large accept a fairly generous social safety net and acknowledge the need for some government regulation for the orderly conduct of business.
Instead, its has drifted towards extreme distrust of the capacity of government to manage the domestic economy summed up in the slogan that it wants to, as Grover Norquist said in a National Public Radio interview in 2001, "drown the government in a bathtub," while retaining support for authoritarian aspects of government such as the military, the Department of Homeland Security and domestic law enforcement. The U.S. GOP is also as skeptical of international trade and diplomatic cooperation, and as hostile to immigration and immigrants as any far right party in Europe.
Moreover, because this faction of the Republican Party managed to seize more or less total control of the party in pre-election coalition building, its leadership owes almost nothing to its economic establishment wing other than tax cuts and the repeal of environmental regulations which it has largely already accomplished at this point. See also here. Since Republicans still need economic establishment support, they have not been quite as disdainful and dismissive of the economic establishment as many European far right parties.
The Republican Party's White Christian nationalist drift has also pushed many non-white and non-Christian demographics such as American Muslims, Evangelical Christian Hispanics, and non-white small business owners out of the Republican Party and into a big tent Democratic Party. the existential issue of their survival drives their political affiliation even though they aren't deeply wedded to many of its left leaning social and economic issue positions. This is a pattern mirrored closely in European politics. But it has been more extreme in the U.S. because the U.S. lacks a center-right party in which conservative non-white and non-christian people can seek refuge when they are unwelcome in a far right party that is openly xenophobic, and often openly racist and openly hateful of non-christians. For example, in October of 2015, a third of likely Republican voters in North Carolina believed that Islam should be illegal in the United States and favored closing mosques in the U.S.
The shifting Democratic party coalition
The Democratic Party has responded to conflict between its factions, mostly choosing moderates and incrementalists to lead it when the chips are down and decisions must be made, recognizing that its progressive wing, while very disgruntled at the state of affairs can't easily defect to the very conservative American Republican Party, and by throwing occasional bones of liberal policy to progressives.
The demise of private sector unions
Unions still side, as they always do everywhere (except for law enforcement unions), with a party of the left. But private sector "blue collar" union membership rates in the U.S. (7%) are lower than when the first significant labor laws were being passed in the early 20th century, while the rate of public sector union membership often "white collar" has surged, weakening the Democrats' ties to the working class. A bare majority of blue collar union members voted Democratic in recent elections. See also here.
From 1930 to 1955 almost all union members were in the private sector. "At the apex of union density in the 1940s, only about 9.8% of public employees were represented by unions, while 33.9% of private, non-agricultural workers had such representation." The public sector also made up a smaller share of the work force in the 1940s than it does today.
The emergence of a divided economic establishment
Some sectors of big business such as the tech industry, the entertainment industry, and globalist business interests have tended to drift towards the Democratic Party, while retail business interests, anti-union employers, many small businesses, and dirty industry has remained within the Republican fold. The financial industry courts both parties.
The Republican Party has increasingly ceded educated professionals to the Democrats in a trade for the Democrats working class, less educated white voters. See, e.g. here.
The emerging gender and generation gaps
The Republican Party has alienated the younger generation and many women (the gender gap between the parties has never been greater), while appealing to men and older voters. This has been driven by increasing Republican cultural conservatism (including comfort with what Democrats perceive as racism and sexism).
Female voter preferences in Presidential elections by year (based upon exit polling):
(Source for last chart).