Just a clarification. I see several mentions of studs. Studded tires aren't the only form of winter tires.
A regular winter tire simply has a softer rubber compound, and a deeper, self clearing tread. The softer rubber is needed in lower temps (-15c) because the rubber of summer or 4 season tires becomes too stiff, and can't form to the texture of the road. Basically, you're not sticking to the road anymore. Imagine the difference between walking on a sidewalk, and a hospital floor.
The deeper tread is needed because 4 season tread clogs up. The tread pattern that's good for channelling water doesn't do dick for snow. You wind up riding up on the snow, instead of biting through to the road surface. In half an inch of snow, that's no big deal. But we're not talking half an inch. Also, new winter tires today have an advanced feature that differienciates them from the old classic off road tire. Thin grooves cut into the lugs, giving up to 20 times the "bite" on ice and snow. Ice tires expand on that technology by making those grooves wavy, adding to the total "edge area". On top of that, ice tires add yet more "bite" by using a foamed rubber compound (Blizzak) or fibers (Toyo).
And for those who want to talk cost, let's talk.
Sure, I have to buy 2 sets of tires for each car. A smart person will even go the added expense of an additional set of rims. But, since the winter tires are only being used in winter, one set will often last the life of the car. And since you're not wearing out your 4 seasons spinning them in winter, they'll last longer too. Total cost of tires ends up the same, over the life of the car. And by paying the cost of the new rims, you save the cost of having to have your tires remounted twice a year. Pays for the rims PDQ. And, it gives you the chance to take those stock rims, and use them for the winters, while you upgrade to a nice shiny set of mags to offset your summer shoes. Something many people are doing anyways. Since noone in their right mind is gonna subject their mags to winter road salt.