>Its the tread pattern/depth not the size of the tyres that makes the difference
Yes and no. Tyre makers' tread patterns vary, as do tread depths. In reality, snow/winter tyres use a different rubber mix from 'ordinary' or 'summer' tyres, and the parts of the tread in contact with the road are moulded differently.
>the ability of the driver
When I still lived 'down south' in a posh area of Kent close to London, some snow had fallen during a January day. Returning home from work, I was using the back lanes to avoid the usual jam through the town around 'rush hour'. Came to a short queue of cars at the bottom of a hill. Waited a few minutes, no movement so I drove to the front; the cause was an expensive Audi A5, obviously with 4-wheel drive as all four wheels were spinning uselessly. I happened to be driving an old Honda CR-V - FWD normally but converted automatically to 4WD in slippery conditions; I simply drove around the Audi and carried on up the hill and home.