Is 64-bit faster than 32-bit? As far as I can tell, no. However, 64-bit will allow you to address more memory and virtual memory. You could build a faster computer with the correct components. You are limited with 32-bit.
x86-64 aka AMD64 aka "x64" (in Microsoft-land only) has, in addition to wider registers, more explicitly named registers (double, IIRC) than IA-32 aka x86. In some applications, this can provide a surprising performance boost since you don't have to deal with register aliasing. Some applications can make pretty decent use of this, and MAME is apparently among them which makes sense: lots of the architectures it's emulating, especially for the more modern, high performance titles, have tons of registers compared to x86.
Now, this has nothing to do with the fact that the architecture is 64-bit. The changes just happened to be made at the same time with x86-64.
There are also some instances where 64-bit code can hurt you. Specifically, all your pointers and register saves (on the stack) are now 64-bits which means they take up twice as much room in memory. Likewise, if you go ahead and bump up the size of "int" to 64-bit (Windows keeps it at 32, Linux goes to 64 by default), those also take up more space. Generally, in the grand scheme of things, this doesn't cause a huge increase in resident set size for an application as most of that is other "assets" like graphics, text strings, and the code itself, which doesn't get much bigger and in fact can get smaller, but it can be a somewhat big increase in terms of keeping things in the CPU's cache, and main memory is of course much slower than cache.
In other words, going from IA-32 (x86) to x86-64 (AMD64/"x64") is not guaranteed to result in a performance difference in a given application, and it may result in either a loss of performance (usually small, at least) or an increase in performance (sometimes very substantial).