Amazon.com has begun selling downloadable DRM-free MP3's. I suspect there is a great deal of overlap between Amazon's selection and the iTunes Store, because which songs are available for download without DRM is determined by the companies that own the rights to them.
So, what's the difference between the two services? As I understand it, right now the Amazon tunes cost a little less, though that could of course change. The bigger issue (IMHO) is the difference in delivery vehicles. Amazon allows buying and downloading single songs directly through your browser (full albums do require installing a small program, but it's supposed to be unobtrusive); managing the songs in their folders is your problem. The iTunes Store uses a fairly large, but also feature-rich, custom client for buying, downloading, and managing your music. I like the simpler approach myself, but I can see how some people might not. (Update: Case in point -- I just realized that while "the simpler approach", for me, means Amazon's system, for others, it might mean iTunes.)