That's a good question. Most of the files used are exactly the same in both modes. The "msvcp" DLLs and the plugins DLLs are different files but their usefulness is limited.
FF9SpecialEffectPlugin.dll is used to render the spell effects (particle emitters and stuff like that) so maybe there's a graphical difference for spells (the other models and backgrounds are the same in x64 and x86).
SdLib.dll is used for the sound, so maybe there's a sound quality difference (in both case, that quality is bad though because of the bad frequency ; there are mods to fix that DLL).
But I don't know anything specific.