You used to buy and then download. Then, the would put the music in your Amazon Cloud, which for me was great, because I used the Cloud Drive all the time and it gave me a backup.
I just downloaded an MP3 album and it never showed up in on the Cloud Drive. I installed the Amazon Cloud Player (which annoyed me) and the songs showed up there. Where are they? I see a download button. But wait, I have install the Amazon MP3 Downloader software. WTF?
I still would rather deal with Amazon than iTunes any day, but come on people. Buy, Download... it should be two clicks.