The answer by @JamesK is very good but misses some key points that I expand upon here.
In a parliamentary system, when, as the Democrats do now, the majority party holds a very thin majority, a vote against the majority party position by a member of that party (or the majority coalition if it is made up of more than one party) on an issue which has not been designated a "free choice vote" results in parliament being dissolved and new elections being held.
Effectively it disempowers members of that representative's own party from exercising legislative or executive branch power in the future until the new election is held and a new coalition government is formed, because political norms call for the prime minister and the prime minister's cabinet to act as a caretaker government pending the outcome of that election (usually held within two or three months of the vote). Failure to support the majority in these system basically shuts down the entire legislative process for several months with no guarantees that the post-election legislature will be better.
Indeed, in situations where the prime minster and the prime minister's cabinet are elected with the support of a coalition of multiple political parties, a failure to a member of the majority coalition to support the prime minister that triggers a snap election isn't uncommon.
This is particularly common in systems like Israel and until recently the system used in Italy, where proportional representation allows many small political parties to form (unlike in Germany where a party needs 5% of the vote to get seats in the parliament) getting a share of the legislature exactly proportional to their share of the vote with seats filled from party lists. In those systems almost all prime ministers require coalition support to control a majority, and those coalitions often includes many parties. Italy and Israel have each had dozens of short legislative sessions due to snap elections triggered by the collapse of a multiparty coalition government when one or more member parties broke ranks. But, because voters don't change much in the year or two between snap elections in these systems, the parties available to form a new coalition are often very similar to those in the government that fell due to lack of majority support on some key legislative issue. Overall, people in those countries find this experience to be very frustrating and it leads to a great deal of cynicism about politicians and the political process beyond the already ample norm for any representative democracy.
Indeed, the early part of the realignment process in the United States, roughly speaking in the late 1970s and 1980s, the U.S. while it nominally had a two party system, in practice, really had a de facto three party system made up of Northern Democrats, Southern Democrats, and Republicans, in which Southern Democrats sided with Northern Democrats to organize Congress procedurally and on economic issues, but sided with Republicans on many social issues and on military affairs. Eventually, however, realignment proceeded further, and the former Southern Democrats were replaced by Democrats from the South that were genuinely liberal (mostly black) and by Republicans from the South who were conservative. Again, however, because the Presidency was left undisturbed unlike a parliamentary system, this issue by issue decision-making as the Southern Democrats sometimes favored one faction and sometimes favored another, worked out without undue disruption to the functioning of the government.
In contrast, a breach of party discipline in in the U.S. legislative process undermines that party's success only on that particular piece of legislation. The majority party remains in procedural control of the chamber and retains a majority which can allow the majority to pass future legislation on other subjects which even moderates in a party prefer more often than the other party's position, until the next scheduled election.
The U.S. does not hold snap elections when the majority party loses support on an issue (or even loses control of the chamber and loses a majority due to a vacancy, a vacancy election, or a party switch by a member of the majority party). The U.S. has experienced changes of partisan control between elections in a chamber.
In particular, a party with a President in power, like Biden for the Democrats now, continues to control the executive branch and the regulatory process even if it can't enact anything but a minimum common denominator set of appropriations bills necessary to keep the government open and functioning, and approves sufficient Presidential political appointees that require Senate consent for the government to continue to function. Congressional norms and exceptions to the filibuster rule in the Senate generally allow this to happen despite deadlock on other matters.
In the U.S., with its bicameral system, at long as the President's party controls either the House or the Senate or has an effective filibuster position in the Senate, the President doesn't have to devote much time to fending off hostile legislation which can't pass anyway, and proceeds without significant new legislation.
Also, it bears noting that the party does not entirely lack "carrots" and "sticks".
A legislative party can award desired committee seats and/or leadership posts to loyal members and deny them to members who are disloyal (although the bargain can also go the other way with a swing voting member of the paper receiving favorable treatment in order to vote in favor of party priorities on some issues despite being, overall disloyal). Indeed, in rare cases, members who have engaged in serious misconduct or disloyalty are stripped of all committee seats and leadership posts, and often have trouble getting re-elected even in a safe district, as a result.
Also, the non-legislative part of the party can choose to supply or deny election campaign resources to members based upon their loyalty in terms of party discipline. Again, however, this isn't always how it works out. Unreliable members of the party with poor party discipline tend to be in districts where the majority party is lucky to have anyone in office at all (such as Senator Manchin, a Democrat from West Virginia which was one of the most pro-Trump states in the United States in the 2016 and 2020 Presidential elections).
If a member of a party is too disloyal to the party line in a district that is reasonable safe for the party, the rank and file primary and caucus voters in that district can frequently support a primary challenger to the incumbent. But even then, this is not done lightly. This is because so much power in Congress (particularly with regard to earmarking spending for one's home district) is allocated on the basis of seniority. It is also because incumbents empirically have such a strong electoral advantage relative to candidates of the same party in an open race in the same district. So disloyal incumbents are really only at risk of a primary in a seat that is safe for that incumbent's party and the harm done by the disloyalty is perceived to outweigh the benefits that the district receives from that incumbent's seniority.
Allowing deviations from the party line is one way that the party makes its members for districts that naturally favor the other main party or are closely divided to win re-election by appearing more moderate than the party as a whole which the district wouldn't support.