i think youtube is designed in a way that it doenst run in the background to keep the music playing. In the past, the versions would let you enjoy the audio while having the screen off (on phones) but not anymore. They want you to keep it on, or at least it's designed that way.
However, if you have a music app, such as Google Music, Xiaa Live, TuneIn Radio etc, you can start to play music, and then "minimize" or change apps and it will continue to play.