The Music app continues to work, but the in-browser apps stop because the OS isn't aware of anything needing to continue playing. There is no official Pandora app that I could find, so if the developer did what most seem to do today and wrap a web browser control and do nothing but show the website, that would cause the same thing to happen. If the dev does it the proper way, it should continue playing after the screen goes to sleep (or pressing the power button). Try MetroRadio for Windows 8, that seems to do what you're expecting.
An easy way to tell is if you leave the app and open another and music continues to play, it should also continue to play when the screen is off. Another way is to press the volume control. If the app is properly built and registers playback with the OS, you should see track information. If not, you'll only see the volume control.
Here's an example:
In this app (Metro Radio), if I press volume, I'll see the track info. If I switch apps by pressing Start, or hit the power button to put the screen to sleep music keeps on playing. That's because the guy who built this isn't a *completely ****ty developer* like the next one.
This app (Pandora Radio Touch or something stupid like that) is just a web browser wrapped in an app. It literally does nothing except for open Internet Explorer without an address bar and only go to Pandora.com (well, that and serve a bunch of ads). Because of this, if I press volume, I get this below (no track information), and if I turn off the screen my music stops. Moral of this story? Browser-based music services won't keep rocking after the screen goes off. ****ty developers who churn out crap apps just annoy people. Don't be a ****ty developer!
