Weight is a small part of winter performance. After all a 100# tire and rim has a 4,000# car pushing down on it so 100#s is not really significant.
What does make the difference is tire width, narrow is better, tread block design(open is better) and rubber compound(soft and squishy is better).
Take a good look at the majority of snow tires, they are usually as narrow or narrower than stock. This is to help the tire cut thru snow and crud and not "float" on top of it. You can find fat "performance" snow tires, but narrow is usually better and easier to find and much cheaper too.
An open tread block design is required so the tire can throw off snow and not pack it into the tread where it acts like ice on ice. Street tires usually have narrow grooves for smooth quiet driving. Open blocks are noisier and don't handle as well on dry pavement.
Soft rubber compound is usually needed so that the tire conforms to the road, some even have tiny bubbles to act like suction cups. Normal summer rubber gets hard and stiff which can make it skid easier. Soft rubber has a negative though, they will wear out faster and most snow tires don't have soft rubber all the way thru. Usually you have 5,000 to maybe 10,000 miles of soft rubber and then under that is normal rubber. So you have a snow tire as long as the soft rubber lasts, then it becomes a "summer" tire with open treads.
One thing you might look into if you can't afford real snow tires is you can have almost any tire Siped. This is tiny little cuts all around the tread which gives the tire many more gripping edges to handle ice and water. On snow you still have the problem of tread design. I never tried it but have heard it works pretty well and doesn't affect normal wear and handling much.
Just stuff I read and from some experience with my own snow driving.
Jim Henderson