With an open differential (what most passenger cars have) the power supplied by the engine will take the path of least resistance. More specifically if one tire has traction and the other does not, all of the engines power will be sent to the slipping tire and not to the tire with traction. Airing down tires in snow, mud, sand, etc, will greatly increase traction by giving the tire a larger surface area on the ground. If you get stuck in a situation where one tire has traction and the other does not, airing down the spinning tire may help to balance out the power distribution a little. However, once you are un-stuck, you should air down the other tire as well. Airing down, in some situations, can almost provide as large of a traction gain as shifting into 4wd.


How dirrentials work: http://auto.howstuffworks.com/differential.htm
Airing down: http://www.californiajeeper.com/tire-presures.htm