It will put music on your phone at your request. To the right of the song or album icon, there is a grey circle. If you push that button, it turns into a safety pin, telling Play to keep those files on the device. If you see the safety pin next to an album or artist icon, just push the button and it will allow the files to be overwritten on your phone.
Play will also cache your recently played music on your phone, depending on how much memory you have. If you want it to stop doing that, you can go into Settings/Automatically cache and uncheck the box. If you want to see what music is on your N4, touch the bar at the top that says All music and select On device only (unless it already says On device).