Every item ever purchased, solves a problem, and it is unclear what problem this one solves.
You need imagination to figure that out...and have used a 3DS/DS
Menus, rather then disturb gameplay to go to another screen for something you just look down.
Play on the controller while someone else watches the TV.
Internet is better to look at up close than on a TV screen.
In put for stuff like drawing etc.
Easy access to maps, you can even draw positions and reminders...something like this can improve many games especially Akrham City.
Easier controls for RTS games.
Better for management sims.
If you're playing a game against a buddy, one can see the view on the TV the other on the controller..to me that's better than split-screen squashovision.
Precise aiming would be seemless, like say you're scoping someone in a shooter you can look down on the controller and then if you hear something you can quickly look up.
Enhances puzzle games.
Rather than having to use a button/menu to scroll for a weapon/object...just tap the screen.
Driving games, have a certain set view on the controller.
Games that need precision like Trauma Team.
Play a RPG, when you come across something you dislike chances are this controller could solve that.
Simulations can have everything they need on the screen for a more immerse experience.
That's just the stuff you can do with the screen, the controller has cameras, so great for video chat on games. See your mates face in the corner of the screen so you can see their expression when you kill them. That's also great for social titles, MMO's...shit MMO's in generally could finally be done properly on a consoles. There are motion controls too built in so again more precise aiming, "reveal" based mechanics, etc etc.
Here's a video that I feel did well in presenting some ideas:
Sony can do a lot of this with the Vita and I really hope they do, would make a lot of games a lot less annoying.