Well, it's not a simple answer. I think the Tesla AWD has much better traction control, but it has to engage more often.
The old Quattro I had was Torsen based, which is the mechanical full-time AWD that was configured on that car to distribute torque 50/50. This is a bad setup for fuel efficiency and performance driving (due to understeer bias), which is why Audi finally ditched it a few years ago for a part-time electronic AWD system. So a recent Audi would be a much better comparison to a Tesla. That said, the FT 50/50 distribution is IMO the gold standard for snow and ice driving.
I believe that the AWD Tesla has the potential to be the best system out there, if I was only able to force the system into 50/50 (or similar) for snow driving. IINM, the Performance cars can do it. The advantage of that arrangement is that I don't have to wait for the car to react to a loss of traction by engaging the front. This is what I occasionally notice in my non-P AWD: the rear will start to just slip or kick a little bit and then the front starts to pull and straighten everything out. I've actually found I can kludge around this a little bit by being more liberal on the accelerator to prematurely engage the front when powering out of an ice covered parking lot, for example. Again, if we could be tossed a small bone here by just getting an on/off or "snow" button which just does the 50/50 until disabled, we'd be gold. IMO this wouldn't cannibalize the flexibility offered in the Performance models.