Many people use air pillows, but some do not. The "real" reason for the pillow is to provide a cushion for ice to expand against if your pool freezes. The thought being if you did not have one, the ice would instead push against your pool walls.
However, many people also find the pillows useful for keeping water off the cover. It makes it easier to drain when there is a "moat" effect. Some people pump off their covers (I used to before mine started leaking) - it lets the leaves blow off. Others let water get on top - it helps with wind. You just want to keep any water at or below the normal water level in the pool when it is full.
There is not much you can do about snow, except look at it. As long as your cover is resting on the water underneath, it should be fine. If you have it stretched like a drum, the weight of the snow is going to rip it. Even in a bad winter, I tend to find a few days of a warm spell where there is meltwater in the cover. I just pump that off.
-dave