How exactly do you define "bad/cold weather"?

How much snow and ice are considering riding on? How much calcium will be on those roads? If any, you may want to consider a "winter bike", i.e. a cheaper bike that you will not be too sad to see quickly deteriorating as you ride through the winter snow-abrasive-calcium slush.

For stability, you'll be fine with tires with a bit more thread to them. I don't think a mountain bike would give you much of an advantage there, unless you plan on starting to jump over snow piles...