If you have an x64 OS, running x64 software can make full usage of all the hardware capabilities. 64 bits programs may be optimized with specific x64 processor instructions that make processing more efficient than x86.
So, in short, using x64 software gives you more performance than the x86 equivalent, and also the ability to handle more memory per program if you have it. Functionally, both should be equivalent.
EDIT: This is not photoshop specific. All that actually applies to any program that has both x86 and x64 versions. Both do the same, but the 64 bits may run faster.