All other things being equal, the A1200 will allow you to run AGA games, which the A500 will not do, no matter how you expand it. However there aren't that many AGA games so it might not matter too much.
The A1200 is also more readily expandable if you want to play games that need (or at least benefit from) a hard drive and extra RAM and such, even if they don't particularly require AGA.
If you mainly want to play older games that come on one or two floppies, an A500 and a floppy drive emulator (so you can run the games from a memory card of some sort) is probably the most cost-effective option.
Generally speaking, apart from the AGA/OCS divide, any Amiga can run any game as long as other requirements are met. And as demolition already mentioned WHDLoad is a great way to smooth out any differences.