Why do you need winter tires? The fast answer is handling. Well-designed winter tires have deeper treads than summer or all-season tires. (The latter, by the way, are really three-season tires if you live in the snow belt.)
Winter tires’ deeper treads help them deal with snow and the icy precipitation that creates slush. An interesting side benefit of winter tires is that they improve traction by packing snow in those treads for better grip on snow.
Also, winter tires are designed with tiny slits in the treads (or as Bridgestone calls them “snipes”). These provide biting edges on ice that help with acceleration, deceleration, and stopping.