TL;DR: It all depends, and you need to benchmark it, but the big differences besides memory are that you have twice as many registers, pointers are twice as big, and if newer software is limited to one type of architecture, it's probably going to be 64-bit (for the time being).
Super User has a nice post on the distinctions between 32-bit and 64-bit operating systems and their effect on performance. Ignoring the part about addressing more memory (as you requested), pointers are twice as big (4 bytes in 32-bit, 8 bytes in 64-bit), so you can store half as many pointers in cache, but you have twice as many registers to work with. If the bottleneck in your computations is register memory, then having twice as many can be a big help, and you may see substantial performance gains (the top Super User answer claims as much as 30% from the poster's own experience).
However, if you look at this presentation on molecular dynamics codes by Axel Köhlmeyer, the answer isn't so clear. He claims that he saw no difference in performance between 32-bit and 64-bit versions of the codes he was using, though I found the graphs he presented confusing because he doesn't seem to actually compare 32-bit and 64-bit versions of his codes.
Wade Spires wrote a report when he was a graduate student at UCF on the differences in performance of the SPEC2000 benchmark on 32-bit and 64-bit operating systems. The 64-bit operating system isn't necessarily more performant overall; there are some tasks where programs execute more quickly on the 32-bit OS.
The best advice is to benchmark, so that you know for sure what the performance gains will be. It requires some work on your part, and time that you may not have for benchmarking. If you choose to benchmark, I recommend using a virtual machine as a testbed for these sorts of comparisons so that you can easily switch between guest operating systems on the same machine while doing the minimum amount of setup and teardown work.