Really, both of these are true. You absolutely need to drive at a sensible speed. That's extremely important. If you're flying through snow and ice, doesn't matter what tires you have on, it's going to be very dangerous. But winter tires do stop far better in icy conditions. As someone else said, they have a soft compound of rubber and grip the ice better. But again, if you drive with common sense, all season tires can do just fine for the overwhelming majority of people.
I just can't agree with people who say their car (FWD) or minivan are horrible in the snow, even with good tires. Any FWD vehicle with decent tires will do fine in most typical snow conditions we encounter.