Recently I've played a couple of board games that use apps as an integral part of play. These are X Com - The Board Game and Alchemists.
In both these the app is used as a form of proxy games master, they maintain some form of shared state that is hidden from the players. However, in other respects the game play is fairly traditional - it would have been possible to make these games without the apps.
Whilst these games may not make the most of a companion app I think there is still plenty of potential. Both these games have fairly complex boards and rule sets which fundamentally the app does little to alleviate. Comparing this to a game like Space Team, which is purely a peer-to-peer app based game, and I think there's a lot of areas to explore.
With some form of peer-to-peer communication between players' apps and the computational power available in devices it should be possible to abstract away a lot of the complexity from rules, boards and counters into the app element and free up the players to concentrate on the game rather than the rules.
It doesn't surprise me that the first games to make use of apps are rather complex. They strike me as games designed by people who like game rules.