Both airlines are providing contest incentives to get people to use their apps. American, for example, is holding the "Mobile Million" contest. Just download and install their app by October 15th, play a few games of sudoku (among other things, the AA app has a built-in sudoku game...), and you could win one of many prizes, including the grand prize of one million AAdvantage miles.
Delta started the war of the contests with their "One App. One Million Miles" contest, which runs through September 29th. The big prize here is one million Delta Sky Miles. You have to be a Sky Miles participant to enter this contest with the app. If you're not, it takes less than a minute to join the program.
Although your ability to use all those frequent flyer miles is subject to availability of reward seats, it's still a heck of a prize, and it costs you nothing to enter either of the contests except a little bit of time.
[Tip to Gadling from @LuxeTiffany]