There is very little difference an "average" user will see between the two. Right now the processor in your computer is optimized to execute both 32 and 64 bit code equally well.
For the programmer, 64 bit allows access to higher memory registers natively, without having to use a "trick" interface provided by the CPU to use ram beyond 4GB.
If you're playing a large online game, it might run better as 64 bit because it needs a large contiguous block of ram.
If you're working on a mailing list with tens of millions of members, it might run better as 64 bit for the same reasons.
Eventually Microsoft and Intel will retire 32 bit, same as they did for 8 and 16 bit. That is a long way off.