The difference between32 and 64 bit is about the system you use to play the game. The (very) old systems only can run the 32-bit version, the 64-bit version is suitable for the pc's built in the last decade at least.
Often newer systems have support for 32-bit programs but.. it is just better to stick to 64-bit if your systems supports it.
It is not related to graphics (quality) but relates to the instruction set of your processor and the amount of supported memory..