Modern winter / snow tires utilize soft compounds that bring high levels of snow and ice traction. This level of grip is offered by having three very important characteristics. The first factor is tread depth. The best snow traction comes from a tall, deep tread pattern that will pack snow in, causing snow on snow contact, and therefore, providing excellent traction. When there's less deep tread depth, the tire is less likely to hold snow. This is why tires designed for muddy conditions don't do as well in snow.
The second factor is tread pattern. A big functional feature is the amount of siping a tire has, as sipes are engineered slits in the tire's tread pattern that come open as the tire rolls over the snow, creating more biting edges. Certain retailers offer a service to cut your tires for more all-season traction, however this service hurts the way the tire handles and also imposes uneven treadwear. Tires are extensively engineered, why would you want to cut up the only connection you have with the pavement?
The final factor is tread compound. And while there is diversity among some of the manufacturers on this topic, one common point is a compound that remains flexible at low temperatures. Michelin and General both utilize a high silica compound that improves traction in wet and slushy conditions, while Bridgestone goes one step further with Tube Multicell technology. Tube Multicell is a compound with many long tube-shaped voids in the tread compound that gives the water somewhere to go as the vehicle passes over snow and ice. Bite particles are also a feature in many tires and on a very small scale, bite particles act like little tiny studs and offer more biting edges for snow grip.
If you're looking for more information on identifying winter / snow tires, read "How to Confirm a Winter Performer