The Republican and the Democratic parties are "broad church" organisations compared to UK parties. There is a large overlap between the memberships on both sides. Republicans differ most from Democrats in how they view the function of the Federal Government. Republicans favour small government - limited "interference" in the lives of ordinary citizens, a reliance on the power of the market and (where possible) low taxation. This can be at odds with a strong defence system and international standing but they have to learn to live with that!
FDR in his first campaign seemed to be a Democrat quoting from the Republican handbook. Hoover was the one proposing using government funds to revitalise the economy. Once in office, however, faced with an economy at rock-bottom, FDR became a firm believer in big government, at least in terms of its interventionist powers. His solution to The Depression was to spend America's way out of trouble. On the other hand he also managed to reduce the Federal budget by cutting pension payments to war widows and veterans. He also was an inteventionist in overseas problems - he made no secret of his views on the desirability of assisting Britain and its allies long prior to Pearl Harbour.
It is also worthwhile remembering the nature of the presidency. Once elected, the party is to some extent 'left behind' in the Senate and the House - very different from the British system where the Executive springs directly from the Commons. FDR's cabinets were all to some extent 'coalition' exercises and some of his fiercest critics were from within the Democratic party - particularly from New York, which he saw as corrupt.
Final point, I assume when you talk of Taft, you mean Robert ( Senator during the FDR era) rather than William (President 1909-1913) . Robert Taft actually formed a strong alliance with many Southern Democrats to severely limit FDR's ability to pass "New Deal" legislation.