main difference is that x64 lets you use more than 3.5GB of RAM. on a 32 bit OS you can't use any more than 3.5GB of RAM but on 64 bit you can use up to something stupid like a few hundred GB, so it is, in personal computing terms, unlimited.
There are also different drivers for 64 bit OS. The majority of software is signed for 32 bit OS but not 64, however most new software is now signed for 64bit aswell because it is becoming more popular with a mid range computer now having 4GB of RAM. Hardware needs different drivers but all hardware that is made now is suitable for 64 bit use. Anything from P4/First dual core procesors onwards can be used with 64 bit operating systems. the AMD athlon 64 is called so because it is workable with 64 bit OS, sometehing novel when it first came out.
Finally the file order is slightly different on 64 bit OS. On Vista for example, you will probably know that in c:\ there is the file of "prgram files" . On 64 bit vista there is "program files" and, the more commonly used by programs, "program files x86" . It puts all programs that have 64 bit and 32 bit versions (most programs do but don't say, it just does it itself in it's programming) in the x86 folder. So you will find that the x86 will look like the program file folder in a 32 bit OS and the "program files" folder will look pretty empty