I used to store my summer tires in my garage during the winter. The garage usually stays above freezing, but the last few years have seen a week or two in which the garage dropped below freezing. I noticed after the summer tires were exposed to freezing temps in storage that they became louder, tread compound was harder, and the level of traction dropped. The tires were about half worn, but became junk. Now I store my tires inside the house. Tire Rack suggests keeping summer tires stored above freezing and now I understand why.
Summer tire compound isn't designed for colder temps. When the compound is exposed to cold temps, the molecules change their alignment. This is why summer tires don't grip well in cold temps. After being exposed to cold temps, the molecules may not return to their proper state, causing them to not work as well in the future. Tires work by the rubber interacting with the road surface. A summer tire can mold itself around the road surface. The softer the tire, the more it can mold to the surface. A harder tire can't mold around the surface, therefore it has less grip. When a soft summer tire is frozen, the molecule alignment changes and the compound can't react as well to the road surface. This is why my summer tires, even though they were never driven below freezing, became useless after being stored in cold temps. Most summer tires start to see molecular changes below 45'F, which is why there is a warning not to operate the tires below the temp.
Moving beyond the compound, the tread design is such that exposure to snow or ice is dangerous. The large tread blocks don't leave enough room on the tire surface to evacuate snow. The large blocks also create a large load dispersion on ice, giving you less traction. What works great on dry pavement is terrible on ice or on snow. It is why ice skates have sharp edges. They concentrate the load over a small area. Force equals load over area. Load is constant, so changing the area increases the force. Winter tires also have tiny cuts all over the tread blocks that allow them to vibrate and clear away snow. This makes them self-evacuating. A summer tire makes one revolution on snow and becomes completely packed. You basically have a race slick made of snow. Some winter tires have multiple compounds in each tread block that allow for increased ice traction. This is from a super sticky compound that is formulated for cold temps. At high temps, it breaks apart and you can destroy an ice tire. At cold temps, it works great.
Finally, sidewall construction of a summer tire is designed for maximum feedback and cornering stiffness. A stiff sidewall increases the wheel rate, much like a stiffer spring does on the suspension. When increasing the wheel rate, the level of warning prior to traction loss is decreased. On snow or ice, the overall level of traction is greatly reduced, therefore having a stiff sidewall provides you with less traction and less warning of traction loss. A winter tire has a softer sidewall designed to lower the wheel rate and provide more traction. The softer sidewall also flexes more causing the tire to warm faster. Going with a smaller diameter wheel with taller sidewall also protects the wheel from damage in case of pot holes or other winter road damage.
I will always discourage all-seasons since they suck in all-seasons. They have to cover to broad of a temp and traction range to be effective. Even on my wife's AWD Escape, we run winter tires in the winter. The all-seasons provided very little traction in braking, terrible corning traction, and were scary on ice. We installed winter tires and the Escape is incredible in the winter. We still use all-seasons in the summer, mostly due to the fact that the factory tires are in such good shape from only being used in the summer. The vehicle's design wouldn't allow for the advantages of a summer tire to be exploited. We use a higher performance all-season that has high levels of wet and dry grip along with being quiet. The trade-off is an all-season that really sucks in the snow, which is why we don't use them in the snow. They are an all-season tire that is really a 3-season tire.
Most people argue that winter tires aren't worth it. My sister was one of those people. She didn't want to spend the money on winter wheels and tires for her Escape. She crashed twice in the snow because the all-seasons couldn't provide enough braking traction to avoid a crash. Two insurance deductibles, increased insurance rates, and then finally buying the right tires.