The only real reason to run the 32-bit version of Photoshop is to access older plugins that haven't been updated to support the 64-bit version.
There may be issues with supporting older hardware too but it's unlikely.
64-bit Photoshop has performance benefits. There is increased processor speeds which speeds up some operations, the maximum amount of RAM Photoshop can access is also increased (the amount varies depending on platform/OS). You can read more on the Adobe website:
Photoshop Help /
64-bit OS benefits, limitations
The primary advantage of using the 64-bit version of Photoshop CS5 is to access amounts of RAM beyond what Photoshop can access as a 32-bit application. Photoshop can take the best advantage of more than 4 GB of RAM only if it runs as a 64-bit application. If you use files that need more than 4 GB of RAM, and you have enough RAM, all processing you perform on large images can be done in RAM. It's not necessary for Photoshop to swap out to the hard disk.