The bottom line is that there probably isn't a perceptible difference for most end users, but 64-bit is the wave of the future (present?) and does offer significant improvements. You'll be able to run lots of apps concurrently with either 32-bit or 64-bit, but with the 64-bit and lots of RAM, performance will be better.
The previous answers have correctly stated that the amount of usable RAM ("addressable memory" in nerd-speak) is a key difference between 32-bit and 64-bit. It should be obvious that your can only run 64-bit code on a 64-bit processor. However, even an OS that is not "fully" 64-bit can run 64-bit processes. For example, OS X has added 64-bit support in stages with 10.4 and 10.5, and now 10.6 ("Snow Leopard") pretty much everything from the kernel to the included applications are 64-bit.
As far as classifying operating systems based on 32/64-bit, it's actually somewhat of a false dichotomy. Windows ships separate 32/64-bit versions, but (for example) OS X ships as a single OS that can run in either mode. The reason Apple can do this is because they invested in Rosetta and also developed and promoted universal binaries. At first this was to make the PowerPC → Intel transition work, but now they can also leverage the technology to bundle 32/64-bit binaries in the same application, and the OS can choose which to launch as necessary. Windows support is likely to be somewhat different.
The vastly increased memory limits aren't really that important to end users (at least not now), but are critical for high-performance and memory-intensive applications. Even so, if a 64-bit machine and OS work for you, it's smart to buy with an eye to the future, because that's where computing is headed.
(Disclaimer: Nerdy details follow...)
One low-level but important detail that has been left out is that the 64-bit Intel (nee AMD) architecture (aka "x86_64") has a different CPU structure that adds some efficiency over the 32-bit architecture. There are twice as many "program registers", which is basically wicked fast memory that the CPU can use to cache information, resulting in a performance gain. Also, 64-bit code and language runtimes can discard years of backward compatibility with 32-bit cruft and make some new optimizations to code and execution pipelining that were previously impossible, and ongoing compiler development will continue to increase the gap.
Edit: See this AppleInsider.com post now that Snow Leopard has been released: http://www.appleinsider.com/articles/09/09/02/inside_mac_os_x_snow_leopard_64_bits.html