All seasons should probably be called "Three Season Tires" for most of the country. They work below freezing, but poorly, and only on dry pavement. Once you ask them to handle snow/ice, their grip drops to somewhat hazardous levels. You're fine with them in warm areas that don't get storms, but on ice... eh.
There are two (general) types of winter tires: Studded, and studless.
Typically (not always) a studded snow tire will have big open tread patterns and carbide studs for traction on ice. They clear snow very well, and are great in loose snow or slush - they clear it out and get you a lot of grip. The studs grip deep ice and hard packed snow very well, but the traction on dry pavement is somewhat poor (the open tread pattern means less contact area, and the rubber is hard and being held up by the studs), and they're hard on roads. Studded tires are basically worthless on black ice or glaze ice (a very thin layer of ice), because the studs can't grip the ice (they just rip through it), and the rubber can't grab the ice either.
Studless tires tend to rely on a soft, sponge-like rubber that wicks water away from the surface and generally grips ice very well. They don't have as open of a tread pattern as the snow tires, so they're not as good in deep snow or slush, but they're still far better than all seasons. They do grip black ice and glaze ice well, because they sponge to it and are able to get good traction against it. On the flip side, they're only good for 3-4 years before the rubber starts getting hard (even if you don't use them much), and the ones I've used have dry pavement handling that's sketchy, at best.
For most people, if you live in a city or suburban area, studless tires are the right answer. Studded snow tires are nice if you're regularly up in the mountains, or in a rural area with poor snow clearing, but for city use, go studless.