Microsoft's website has an article covering the differences in more detail.
Programs written without explicit 64 bit support should work on 64 bit Windows, since programs in the Program Files (x86) directory are emulated. If you want to know beforehand whether an old version of a program works in 64 bit Windows, you could look at the system requirements page in the Wayback Machine.
Since Windows 7 at both 32 bit and 64 bit are sold together, you can always reinstall with 32 bit Windows if it turns out a program you really need only works on the 32 bit version.
1 Some CPUs support this feature in 32 bit windows through PAE (as said by grawity).