If you hunt small game or are in a survival situation, an air rifle can be an excellent tool. Birds, squirrels, rabbits and other small game can be easily harvested, even with BBs, and pellets can be used to hunt larger animals. I don't eat meat, but my dogs do, and in an emergency situation I would feed my pets and family one way or another.
Of course, lower power means you need to have more accuracy to make clean kills, often with head shots. This isn't a big problem though, as even the cheap models are pretty easy to learn to shoot with precision. Another advantage is the weight - even loaded down with BBs, these guns weigh nothing compared to a real rifle, so they can be easily hauled around and quickly maneuvered.
Since the rifle is practically silent, game will not be scared away, allowing for multiple shots or more kills. Plus, the silence could be very useful in a more populated area, where you may not want to let people know where you are or what you are doing. This is also an advantage when shooting in a garage or basement, if there are people or animals around who would be disturbed by the noise of gunfire.
Of course, I am just talking about standard .177 caliber air guns here; you can buy all kinds of very expensive and very powerful air rifles that shoot pellets in higher calibers, from .22 to .45. These guns are cool and have advantages of their own, but you can get a brand new Daisy or Crosman for under $50, or as low as 10 bucks if you find one used. If you can find an older model in a yard sale, it may have been made with more durable materials anyway. In a TEOTWAWKI situation, a pump action rifle would be most useful, as you would never need to find compressed air or CO2 canisters to fire it.