The primary one was the policy direction that the party wanted to go in. One of the main reasons that Democrats and Republicans from different states had such radically different positions was that politics prior to the Second World War was generally taking place at the state and local level, and as a result, the Republicans (and Democrats) from different states would have wildly different policy positions. As politics in the 20th century increasingly shifted to the national level, there was increasing pressure on both parties to have an ideologically consistent position.

The Democrats had done most of the legwork in unifying the party in the 30s and 40s, mostly by the New Deal. Socially conservative and segregationist Democrats agreed to vote in line with their more socially liberal counterparts with respect to things like welfare and foreign policy, and in exchange the more socially liberal Democrats would deliberately avoid topics like segregation at the federal level. It wasn't exactly a formal arrangement, but more of an understanding between people who had some common policy goals.

The Republicans, on the other hand, were an absolute wreck of a party ideologically speaking. Previously, one of the few things that unified them was opposition to the New Deal. By the 50s the party was split between the more liberal, internationalist, pro-New Deal wing [e.g. Dwight Eisenhower, Richard Nixon, and Nelson Rockefeller], and the more conservative, isolationist, anti-New Deal wing [e.g. Barry Goldwater, Robert Taft, and John W. Bricker]. Eisenhower was repeatedly under attack from the conservative wing of his own party for failing to challenge any of Roosevelt-Truman domestic policy reforms, and for a perceived failure to challenge the Democrats on budget proposals with high price tags. While the Democrats had mostly agreed to keep quiet about the things they disagreed about, the Republicans were openly fighting about the Eisenhower administration's policies.

William F. Buckley Jr. founded the conservative magazine National Review in 1955, and its mission statement is worth reading to give you a sense of why so many conservatives in the Republican party hated the Eisenhower administration as much as they did [it's noteworthy that the magazine refused to endorse Eisenhower in 1956, and refused to endorse Nixon in either 1968 or 1972]. It also contains a statement of principles. An excerpt:

Conservatives in this country — at least those who have not made their peace with the New Deal, and there is serious question whether there are others — are non-licensed nonconformists; and this is dangerous business in a Liberal world, as every editor of this magazine can readily show by pointing to his scars. Radical conservatives in this country have an interesting time of it, for when they are not being suppressed or mutilated by the Liberals, they are being ignored or humiliated by a great many of those of the well-fed Right, whose ignorance and amorality have never been exaggerated for the same reason that one cannot exaggerate infinity.