Cars with 4x4 don't stop any better than a car without 4x4.
That's a common and, usually, quite false quip. All my 4WD vehicles stop better in slick conditions in 4WD vs 2WD, because the front and rear are locked together, which means the front brakes can slow the rear wheels as well, whereas the rear wheels won't be doing much in 2WD for braking in slick conditions. On the flip side, if you lock the wheels up, you lock up all 4 and lose all directional control instead of just locking up the front and turning downhill. So there's that...
On most vehicles, the rear brakes aren't doing much at low brake line pressures. The design is that the front wheels lock before the rears so you still have directional control (fronts locked, rears rolling will keep the car heading forward, fronts rolling and rears locked will try to swap ends, violently, and all 4 locked is zero control and you'll probably slide downhill and rotate a bit). Try it. Lock up the fronts on ice, I can guarantee your rears will still be rolling.
With 4WD engaged, the whole drivetrain rotates together, so the rears will slow down as well. Since there's almost no weight transfer in slick conditions, they have plenty of traction as well, and can slow the vehicle significantly more than they usually do in slick conditions. You just have to be careful to not lock everything up (it's super obvious when this happens, so just release a bit of pressure, though I think modern antilocks solve a lot of this).
For a maximum effort stop in snow, modern antilock systems may be a bit better about letting all four use full traction without 4WD, but I don't own a vehicle that has 4WD and modern antilocks, so it hasn't been something I've tried.
On important thing to remember . . . you can save a lot of money sometimes by sizing down your tires when getting winter tires. If you have 17 inch rims, try getting 16s with a bigger tire (so the wheel diameter is equivalent). There is no benefit at all to having a large wheel in the winter. Smaller wheels have cheaper tires as a general rule. More rubber means you get more padding/cushioning, you can run them at lower pressures (which means that they'll need less topping up because less air will leak out), and thinner tires mean that they'll punch through snow better.
The only thing you have to keep in mind is brake clearance - on some cars, smaller wheels simply don't fit. So check that before you rely on downsizing.
But, you definitely want the thinnest winter tires you can find. The idea would be pizza cutters. The difference in snow penetration between 185s and 165s is very noticeable - I've run both in the past, and the 165s were a lot better.