I like being able to listen to music when I have no internet connection.
I like being able to listen to something when the original was taken down.
I like being able to listen to something that the vendor no longer offers.
I like having smart playlists in my library and on my devices that automatically update themselves based on meta data such as what I've played recently.
Unfortunately I like being able to use my music program on an operating system of my choosing as well, and am looking for an iTunes replacement. But then replacement can't be worse than the original. That's a deal breaker.
Amazon doesn't just provide a link to the files once you purchase them, there is a downloader client. I have never tried the official one, there is an open-source command line alternative called "clamz" that works perfectly.
Personally I usually have an internet connection except on the train, so I use a Subsonic server, and my Android Subsonic client is set to aggressively precache my playlist (so it plays happily through tunnels, even if the tunnel lasts 10+ songs).