The shark in water but the pitbull on landIs this one of these "Which would win out of a Pitbull and a Shark" questions?
I think if there was a knife fight, with a wide tyre versus a narrow tyre, the wide tyre would win.
If there were no knives or guns, it would be pretty close but the wide tyre would still edge it.
On snow and ice you want tyres as thin as possible. This reduces your contact patch and increases the pressure on the contact area which will help create more traction, also a long thin contact area will have more directional stability than a short fat one. Most studded tyres are fatter but that is to allow room for the studs. Don't believe me, compare WRC cars on their normal tyres:
and then them running in the swedish rally, the tyres look comically thin compared to normal.