[1] http://www.baaqii.com/promanage/productimage/Ewhole/A/A678-1...
One nice thing about the phone apps is that you can search for a different thing to watch _while_ currently watching something. Fast-forward/rewind also tends to be a little more intuitive with adraggable progress bar, except small-scale rewinds (1-2 minutes) of a movie is _harder_ because a few pixels is difficult to select cleanly.
I will say though that Prime is _freaking fantastic_ when streamed from my phone, better even than using the native TV interface. The Prime phone app lets you navigate _by scene_ in a show (e.g. Bosch) which my tv app won't let me do. I wish Netflix and D+ would/could copy that, specifically. I wish it didn't have spoilers in scene descriptions, but being able to say, "Hey go back to the start of the council of Elrond" is really nice, rather than having to use +/- 30 second increments.
But when you cast from an application with integrated chromecast support (as e.g. Netflix) then the phone is free to use as you wish while the kids are watching. I do this all the time. I start the cast, then use my normal TV remote to pause, for volume control etc. You don't need the actual app for most things and you can use the phone/tablet for other things (I have to find translations for my wife, for example, while watching).