Mostly, it is a function of how the app is written - and also, it depends on the permissions provided by Apple. There are a lot of apps that will let you re-start them after you've closed them out or gone to another app.
I was going to say that you can re-activate the [iPhone-only] app (after closing it) by doing the four-finger swipe up to reveal the multi-tasking bar (double-clicking the Home button does the same thing), scrolling to the right to reveal the player buttons from the left and pressing the Play button. Except, you can't. The YouTube icon isn't even showing as the app to play.
So, then I tried it with the YouTube player app Jasmine and the process I described works fine - the app continues to play after I press the Play button (and it was showing as the last music app available).
This was all on my Mini, by the way - which is running iOS 6.0.
So, I'm going to go with the statement that it's the developer's and Apple decision as to how the app behaves. If you can't get the app to start playing the music by doing the procedure I described in paragraph two, then it's the way the app was designed.
Hope some of this helped.
Marilyn
