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The My Music app is Motorola's stock music app on the phone. It ties into MotoCast and reads music from your computer. You can download the music to your phone if you always want access to it. Otherwise your computer has to be on for music to stream.
Play Music is the Google music app that now comes loaded on the phone. It works mostly the same way, except your computer doesn't have to be on to stream because you're using Google's cloud. This app also ties into the Google Play Store, where you can download new music.
Personally, I like Google Music more, but I use Motorola's app from time to time. It's really a matter of preference.