In general, a MTB will be able to run larger tires at lower pressures and will be better for traction. The trade off is that they will be considerably slower. My commute is about 20 miles one way and MTB tires add a lot of ride time.
In past winters, I used my road bike with 28c tires without issue. This year I used a cross-bike with 38c tires and disc brakes. I can ride through 3" of fresh snow with no-issues. The problems come after traffic has its way and compacts the snow and turns it to slush. If conditions are really icy, studded tires are a good option.
Brakes are another consideration. Disc brakes work much better than rim brakes in the snow. Rim brakes often jam full of snow and Ice becoming worthless.