I know the obvious answer of Nationalism but I'm looking for a more detailed, comprehensive answer of the differences between the two.
It's hard to pinpoint specific differences if you assume generic ideology as conceived by Mussolini, and compare it to generic ideology umbrella broadly labeled "socialism" - in part because Mussolini was a "reformed" socialist and brought a lot of socialist influences into his thought.
The main difference is that socialism is an economic system (that naturally tended to gravitate towards specific totalitarian political structures when applied in practice due to its nature times human nature), and another one is an ideological system (which even easier gravitates towards specific totalitarian political structures due to underlying ideology, and can arguably be asserted to be incompatible with anything but authoritarian politics).
In theory, you could construct a socialist society which politically isn't authoritarian/totalitarian, at least in small scale - although it would not stably work in practice in a large self-contained country assuming pure socialism, especially of Marxist variety.
On the opposite end, fascism does not directly oppose private ownership of property/means of production as a whole, although under such regimes a lot of industries ended up either wholly nationalized, or under very strict state control although in theory privately owned. Mussolini explicitly noted that the only feasible implementation of socialism was corporatism, as noted in his quotes in my answer here comparing Communism to Fascism.
Adding to that difference is vagueness.
There's no firm, narrow, definition of what socialism is (look at Wikipedia's definition - it's basically "social ownership of means of production" - without even defining how said ownership is implemented).
As such, a rather wide swath of things can be labeled "socialist" (including, though incorrectly, blended mostly-capitalist societies like modern Nordic system countries).
By contrast, Mussolini's system was fairly firmly specified, bust still rather vague (compared with, for example, Nazism which was far more narrow and rigid).
Please note that the main typical critique of differences, often from the left side of the political spectrum, is "Nationalism".
However, we run smack into that previous issue of vagueness. Yes, socialism in theory, as a general vague idea, does not imply nationalism, and some forms of it explicitly oppose nationalism (the word "Internationale" does have a reason to exist").
But the most common implementations of socialism at a country level (USSR, Cuba, China, Iraq) ended up fiercely nationalistic in practice.
On a similar note, a typical answer of racial prejudices of various sorts aren't correct either - the original Italian version of Mussolini's ideology didn't have a strong racial superiority tones, and only later morphed under influence of Germany; whereas again mainline Marxist states ended up either generally racist, or at the very least heavily antisemitic .
I am tempted to say, "socialist is something you call yourself, fascist is something you call others".
But realistically, "socialism" covers very different political and/or social realities and proposals (even, of course, the paradigmatic fascism, which styled itself "national socialism"), which cannot be classified under the same taxon unless we intend a taxonomy that is utterly meaningless.
Scandinavian social democracy styles itself socialism; so did Stalinist Russia or Cambodia under the Khmer Rouge; so does Chinese capitalism. Anarchists are socialists, so are Marxists - and all the subdivisions within them, anarcho-primitivists, anarchocommunists, anarcho-syndicalists, Maoists, Trotskyists, Titoists, and go figure out what sub-subdivisions and sub-sub-subdivisions - Morenists, Pabloites, MIMites, et caterva. Which brings to mind a quote by Ariano Suassuna - "the left divides itself around ideas, the right unites around interests".
So it would be necessary to either narrow down the concept of socialism - so that we define that Scandinavian social-democracy is socialism, but Stalinism is not, or conversely - or to find something that somehow unites the Communist Party of China to anarchist communes, and voters of Jeremy Corbin to followers of Amedeo Bordiga.
But anyway, socialism, whether we narrow its meaning down to smaller subsets of people and organisations, or whether we make its meaning shallow enough to accomodate all those sundry groups, is about ideas. It is an ideology, or a set of loosely connected ideologies.
And I would argue that fascism is not about ideas, and that it is not an ideology, even though some ideas, and some ideological elements, may be of importance to fascists and fascism. I would say that fascism is a practice: more important than what fascists believe, is what they do. And what they do is the following: they seek to restore order through disorder, to enforce law through crime, to restore morality through obscenity. Fascism is always a movement, and a movement to restore something that is deemed lost due to modern ways; but it does seek to restore the past by actively engaging in the practices of the present. Thus they emulate some characteristics of their enemies (Hitler was explicit on this point: it would be impossible to fight socialism/the left/bolshevism with the old fashioned conservative and reactionary parties of the traditional German Right: to have a chance to succeed, fascism would have to organise in ways similar to those of the Communists and Socialdemocrats).
For these reasons, fascism is always engaged in what is called "politics of ambiguity": they emulate the organisation of the Bolsheviks to better defeat the Bolsheviks; they recruit heavily among the lumpenproletariat - Horst Wessel was a pimp - but claim to represent morality against the decay of costums, and law against crime; they denounce the "plutocracy" but are fine with being financed by plutocrats; and they engage in politics in order to destroy politics. Such elements are absent from most non-fascist political movements, be them conservative, liberal, or - and here we can point to your question - socialist. Those either seek to restore old orders or maintain the existing one through instruments compatible with that order, or to destroy the existing order altogether; only fascists seek to restore an older order through - what they call - a revolution.
 When it comes to ideas and ideology, historically all fascisms have depended heavily on nationalism, sexism, and a cult of abstract violence (violence is good in itself, not as an instrument to attain a given goal, but as mode of life).
But this might be a coincidence, or even the result of bias: the two classical fascisms, those of Italy and Germany, were built upon national resentment in frustrated Great Powers (as was the case with the Japanese right-wing dictatorship of the same period), and so it is natural that they would rely heavily on nationalism and belicism, the latter being inseparable from sexism and violence. In the limit case of Spain, where there was no realistic base for Great Power delusions, most of the belicism - anyway, certainly military expansionism (at least within Europe) - was superfluous, and because this characteristic is lacking, we exclude it from the category of fascism.
Socialism is an economic system. It does not take a view on social issues, except where apportioning of financial resources is used for the social good. It is not a form of government either. Socialism can be mixed with liberalism or conservatism, and mixed with Democracy or Authoritarianism to form a complete set of platforms for a national government. Maoist China and the USSR are often thought of as "Communist," or "Socialist," but they can more accurately be categorized as "Authoritarian Socialism."
Fascism has a component that deals with form of government, in that it prescribes an authoritarian state. Fascism is both anti-capitalist and anti-socialist; its economic system deals with the idea that all value should go towards the state, not to corporations or people. This is called "State Socialism" or "National Socialism," but despite the similar sounding names, these are not classic examples of Socialism. All loyalty should be to the state. Fascism is totalitarian, so it prescribes that propaganda be used to control thought and maintain loyalty to the state. Some scholars belief that structural violence (e.g. racism) is a fundamental trait of Fascism to unite a majority of citizens, but this belief is not universally accepted.
Fascist economic ideology supported the profit motive, but emphasized that industries must uphold the national interest as superior to private profit.
Hitler stated that the Nazi Party supported bodenständigen Kapitalismus ("productive capitalism") that was based upon profit earned from one's own labour
Fascists criticized egalitarianism as preserving the weak, and they instead promoted social Darwinist views and policies. They were in principle opposed to the idea of social welfare, arguing that it "encouraged the preservation of the degenerate and the feeble."
[Fascists] favored corporatism and class collaboration, believing that the existence of inequality and social hierarchy was beneficial (contrary to the views of socialists)
Fascist governments encouraged the pursuit of private profit and offered many benefits to large businesses, but they demanded in return that all economic activity should serve the national interest.
To summarize, Socialism only deals with economics, not social issues, nor government systems. Fascism deals with economic platforms, social platforms, and a system of government. On the one system that they are both concerned with, which is economics, they disagree.
Generally Socialism philosophically yearns to achieve social equality, and an elimination of the competitiveness and violence that it believes is inherit in Capitalism. Nazism (a subtype of Fascism) does not seek to achieve social equality, because it seeks a socially hierarchical society, with some racial demographics forced into underclasses. These two concepts are philosophically 100% different.
Since I've now claimed that Nazism is a subtype of Fascism, here are a few quotes from wikipedia's pages on Nazism and Fascism to support that claim.
Nazism is a form of fascism and showed that ideology's disdain for liberal democracy and the parliamentary system, but also incorporated fervent antisemitism, scientific racism, and eugenics into its creed.
Fascism was a major influence on Nazism.
Hitler presented the Nazis as a form of German fascism.
Roderick Stackelberg places fascism—including Nazism, which he says is "a radical variant of fascism"—on the political right
One early admirer of the Italian Fascists was Adolf Hitler, who less than a month after the March had begun to model himself and the Nazi Party upon Mussolini and the Fascists.
In Germany, it contributed to the rise of the National Socialist German Workers' Party, which resulted in the demise of the Weimar Republic and the establishment of the fascist regime, Nazi Germany, under the leadership of Adolf Hitler.
Here are some quotes from wikipedia which illustrate the fact that Fascism (and its subtype Nazism) are not Socialist and are opposed to Socialism and Marxism. Some quotes additionally will depict actual historical fighting between Fascists and Socialists in Italy.
Opposed to liberalism, Marxism and anarchism, fascism is usually placed on the far-right within the traditional left–right spectrum.
One common definition of the term focuses on three concepts: the fascist negations (anti-liberalism, anti-communism and anti-conservatism)
fascism—especially once in power—has historically attacked communism, conservatism and parliamentary liberalism, attracting support primarily from the far-right.
it manifested itself in elite-led but populist "armed party" politics opposing socialism and liberalism
Italian national syndicalists held a common set of principles: the rejection of bourgeois values, democracy, liberalism, Marxism, internationalism and pacifism
The Nazis were strongly influenced by the post–World War I far-right in Germany, which held common beliefs such as anti-Marxism, anti-liberalism and antisemitism, along with nationalism
Chamberlain used his thesis to promote monarchical conservatism while denouncing democracy, liberalism and socialism.
Corradini's views were part of a wider set of perceptions within the right-wing Italian Nationalist Association (ANI), which claimed that Italy's economic backwardness was caused by corruption in its political class, liberalism, and division caused by 'ignoble socialism'.
Fascists identified their primary opponents as the majority of socialists on the left who had opposed intervention in World War I.
The Fascists assisted the anti-socialist campaign by allying with the other parties and the conservative right in a mutual effort to destroy the Italian Socialist Party and labour organizations committed to class identity above national identity.
Beginning in 1922, Fascist paramilitaries escalated their strategy from one of attacking socialist offices and homes of socialist leadership figures to one of violent occupation of cities.
The Fascists attacked the headquarters of socialist and Catholic labour unions in Cremona and imposed forced Italianization upon the German-speaking population of Trent and Bolzano.
In the aftermath of the election, a crisis and political scandal erupted after Socialist Party deputy Giacomo Matteotti was kidnapped and murdered by a Fascist.
Fascism presented itself as a third position, alternative to both international socialism and free market capitalism.
While fascism opposed mainstream socialism, it sometimes regarded itself as a type of nationalist 'socialism' to highlight their commitment to national solidarity and unity.
While fascism denounced the mainstream internationalist and Marxist socialisms, it claimed to economically represent a type of nationalist productivist socialism that while condemning parasitical capitalism, it was willing to accommodate productivist capitalism within it.
They were in principle opposed to the idea of social welfare, arguing that it 'encouraged the preservation of the degenerate and the feeble.'
Fascism denounced Marxism for its advocacy of materialist internationalist class identity