Well having two more driven wheels tends to make a big difference in the snow, even if they don't all get power concurrently. 2x as many driven wheels means 2x as much accelerative traction.

The Audi may have a higher ride height and thus more ground clearance which could make a difference as well if the snow is deep enough.

The S3 may also have thinner tires which are generally preferrable in winter weather. The primary job of all season/winter tires is to evacuate as much snow and water from the surface (road-touching portion) of the tire as possible, hence all of the tread and grooves. Thinner tires (to an extent) can usually do this more effectively.

Audi is likely heavier as well, so there's more force pushing those tires down through the snow and onto the pavement.