For snow, AT tyres.
Snow is like sand, in that, if you claw away the surface it's very loose underneath and will just crumble. If you push down hard on the surface it'll hold and you'll move forward. So, a tyre which doesn't dig at the surface, rather rides over is the best option. You'll get miles further on snow with AT's than you will with MT's. With deep deep snow, momentum is the key, not tread pattern.