UPDATE 6/6/11 @ 11:30am EDT - Google Music is limited to 20,000 songs.
I received my Google Music invitation this morning and immediately signed up. Google Music is a cloud based music service. Once you upload your music files they are accessible from any internet connected computer. There is also a Android app for smartphones and tablets. Amazon also has a cloud music service called Cloud Player. The files are kept in the Cloud Drive. It also has an Android app. I signed up for it when it first became available a couple of months ago. Although the overall concept is awesome - the ability to access your music 24/7 from any web-enabled device without having to access a hard drive and making the storage of music on your iPod or smartphone obsolete - the two services are a bit different.
Amazon Cloud Drive allows for storing all types of files. You get 5GB of free storage, and if you buy an album from the Amazon digital music store, you get an upgrade to 20GB free for one year from date of purchase. You can purchase additional storage at $1 a GB up to 1k. The cloud drive is separate from the cloud player. However, they are linked-your music files stored in your cloud drive are accessible in the cloud player. Amazon wants you to purchase from their digital music store and the incentive is music purchased from it doesn't count against your storage space, which is nice. The cloud player allows for all digital music store purchases to automatically be saved to your cloud drive so you can listen immediately after purchase. One disadvantage is that new music not purchased from the mp3 store is not automatically added to the cloud drive. You have to upload it manually. In order to upload music to the cloud, the free "Amazon MP3 Uploader" app has to be installed on your computer. The cloud drive and player are free. All you need is an Amazon account.
Google Music is currently in beta and invite only, but should be available to all Google users in a few months. It's very similar to the Amazon service but with three key differences. First, Google has free but limited storage of music files. The limit is 20,000 songs. Although there is a limit, it is substantial. I doubt there are many people with a digital music collection containing greater than 20k songs. Second, Google doesn't have a digital music store. Third, Google's service is platform independent and allows for automatic upload of music files from Windows Media Player or iTunes, so whenever you buy new music, it's automatically added to Google Music. It's part of the Google suite of cloud apps so if you have a Google account, you're good to go.
The Winner: Google Music by a narrow margin.
P.S. Apple has a cloud based music service coming out on June 6 called iCloud.