Roosevelt's biggest contribution to American politics was the establishment of the New Deal Coalition. Roosevelt united the traditional Democratic base (conservative Southerners, northern immigrants and Catholics, liberal northerners) along with expanding into traditionally Republican groups (such as blacks and Midwesterners) by promising an end to the Great Depression. The New Deal brought electricity, modernization, jobs, and most importantly, hope, to the South, the Midwest, and America at large. Basically, Roosevelt created common cause by improving people's lives across the board.
The tradition of the South voting Democrat (no matter what) for most of the party's history, boosted by the New Deal Coalition, made it so Southerners kept voting Democrat as long as they held a prominent position in the Democratic Party, and as long as those northeastern elites turned a blind eye to race relations. Despite being a New Yorker, Roosevelt was still 'their guy,' in the same way the northeastern elitist Mitt Romney was 'the guy' for the South in 2012.
The differences began when Kennedy didn't turn a blind eye. He didn't promise total desgregation or anything, but he discussed civil rights enough that it made the Southern base upset. It's somewhat inaccurate to compare Roosevelt and Kennedy for relations with the South, as in 1960, Kennedy had one of the worst performances of a Democrat in the Southern states, losing Virginia, Florida, Kentucky, and Tennessee to Nixon, and Alabama and Mississippi to Dixiecrats.