I'm basing this answer on J.Samuel Walker's book 'Prompt and Utter Destruction: Truman and the Use of Atomic Bombs Against Japan' (revised edition of 2005, ISBN 080-785-60-79). I'm indebted to Alex Wellerstein for pointing out the book in his blog.
First off, use of the bomb against Germany was impossible unless the Nazis could achieve a military miracle of stopping the Allies and/or the Soviets and thereby prolong the war until July 1945. The uranium gun-type bomb (Little Boy) was due on August 1, and the plutonium bomb (Fat Man) was scheduled to be ready for testing in early July. The schedule was dictated by the accumulation of fissile material and could not be significantly sped up with the technology at Manhattan Project's disposal at the time (much more efficient Steenbeck-Zippe centrifuges were a German post-war invention).
Second, it was an invariant of FDR's and Truman's policy to minimize the cost of war as far as American casualties are concerned ('Prompt and Utter Destruction', pp.9-10).
Third, the Interim Committee assembled on May 9, 1945 (Henry L. Stimson (Secretary of War), George L. Harrison (aide to Secretary of War), James F. Byrnes (designated Secretary of State), William L. Clayton (Assistant Secretary of State), Compton/Bush/Conant (the Manhattan Project scientific oversight team)) and George C.Marshall (Chief of Staff, United States Army), Leslie Groves, Robert Oppenheimer and others did not discuss whether the bomb should or should not be used AT ALL ('Prompt and Utter Destruction', p.14).
It is very hard to discern what FDR could or would have done differently had he been alive for a few more months, but we can safely conclude that the A-bomb was yet another weapon, and that this weapon was presumed to inflict great material and psychological damage on the enemy. The technical decision-making/targeting would still have to be devolved to a committee, and it is unwise to think its membership would have been any different from that of the actual Interim Committee.
The political ramifications of the bomb were much more complicated (I'm opening a can of worms, please bear with me):
- the Pacific war could have been ended earlier (the Japanese were ready to negotiate - it is though highly doubtful FDR was capable of a compromise with them);
- FDR/Harry Hopkins could be less heavy-handed in their relations with the Soviets and thus not as reliant on the bomb to intimidate Uncle Joe as Truman;
- the military establishment (and politicians alike) considered razing enemy cities entirely acceptable before and after FDR's death, so there's not much difference here.
Just my $.02, YMMV.