I have both, and before Google Music came out, I used AmazonMP3 exclusively (going so far as to remove all local music from my phone.
Then I got my MusicBeta invite, and now the only time I Use AmazonMP3 is if I'm buying new music.
For me, Google Music had some significant advantages of Amazon.
1. It's optimized for my tablet. I know this might me minor for some people, and it wasn't hugely important to me, but better eye candy is better eye candy.
2. The ability to Pin music is a really useful feature. This is sort of like downloading (allows you to access content while offline) but the difference is that when you go to swap out your offline music, you can remove it from your device just by unpinning it. Another cool feature is if you have a playlist you have pinned to your device, adding a song to that playlist will automatically download it for offline playback.
3. If you have a "Best Of" Album from various artists, when you select the album from ANY of the artists, the entire album will show up. With my ipod, I would have to go to album view to get this, so it's a nice change.
4. This could just be for me, but I think it skips less when you're in a low signal (aka 1xRTT) area, though they are both great over wifi and even Verizon 3g.