I found Google Music to be garbage on my Note 3. I would play my music library (an assortment of music from Music Match Jukebox to Napster and iTunes, and even some tracks that I ripped from old records) and it seems like every 3rd or 4th track it would error out telling me that the file couldn't be played. I think the Google app was bugging out because I didn't have the correct digital rights for some tracks. Instead of the application going to the next song, it would just sit. I couldn't skip to the next song nor click play from my wireless earphones which made it a PITA when I'd have to stop in the middle of my workout, get my phone, and manually advance the playback.
I started using the native Music player and haven't looked back. I also noticed that apparently Google Music was only playing a small section of my music library where the native app seems to hit all my songs. It's almost as if I've found my lost music collection. Yeah, screw Google music player....