For me, tires made a world of difference...
Wide MTB tires are good for tractoring through snow and car snot/car vomit.
Studs are good for gripping ice and other unsavory stuff that's below snow, ice, and car excrement.
This winter, I had a very successful experiment with another device that's super cheap...homemade tire chains! It's the same principle as car tire chains, but on a bike.
For my next winter build, I'm going to make a fixed gear MTB; opinions vary, but I find that fixed gear gives a lot more traction on snow and ice.