If you look at places like Tire Rack, most Summer tires are not recommended for use below 40F. In fact, some of the softer, stickier summer tires have recommendations to not even be stored at temps below freezing.
Winter tires have a compound that performs well in cold temps, with better traction in the dry, wet, slushy, or icy conditions.
Some handle large amounts of snow better than others. It can be a confusing set of choices, but there are tires that work better in heavier snow. Others work better on ice. Others are just better-performing overall in cold, wet conditions.
We don't get heavy snow on the roads here. At worse, I drive two blocks in snow (if it hasn't been plowed in the subdivision), then I hit a state highway, which is always one of the first ones plowed. Winter tires then work well in the wet & slush.
I had Hankook Icebears on the LGT. More of a high-performance winter tire, but handled reasonable amounts of snow just fine (being lowered on coilovers was more of an issue in snow over 4-5"!!).
I now have Blizzaks for my Focus SVT. Got them from the PO, who was very happy with them. I bought the car in March, but we never had any more snow after February this year.