There might be some other answer but...
It seems to be a little bit complicated. If playback is totally stopped, you can play a playlist by pressing the keyboard shortcut for play/pause, which is control-p . But if something is playing or paused, then that won't work. It will just play/pause whatever is already playing.
Pressing Stop first seems to work. Stop, then Play on the playlist you want. If you are using a Media Center Remote (or similar) that might be convenient.
Brian.