Quit Apps While Using AirPlay
Apple's Airplay uses a mirroring model. This is different than Google Chromecast's redirect model. 

When you play Hulu on your iPhone, the iPhone supplies the video to your Apple TV.  As you noticed, it's the iPhone doing the work. That's why you can't quit the app on your iPhone. 

Apple can play video itself from just about all the same sources as your iPhone. The apps need to stay active on your iPhone when using Airplay because they don't hand-off that job to the Apple TV. To get around this, just use the apps in your Apple TV instead. 

Google's use of redirecting the task to the Chromecast also allows Google the ability to insert itself as the 'man in the middle' and see your viewing habits and shows so it can sell more advertising and monitize your use. Apple doesn't do this with Airplay.