It doesn’t snow much in Vancouver, so why would you need snow tires?
The first thing to change in your thinking is the description of the tire. Ten years ago tire manufacturers made snow tires, with heavy tread to grip in snow. Now they make winter tires, which still work well in snow, but work amazingly well in the very common cold rain we get here.
The change is not in the tread pattern, but in the compounds used. Winter tires stay soft in cold weather and grip the road far better than All Season tires (which should really be thought of as 2.5 season tires here).
From late fall to early spring the difference is very noticeable. You can get by with the All Seasons, but all it takes is one need for a sudden move or braking on a wet cold day and the winter tires will have paid for themselves. This is particularly important with the wide performance tires that are now common on many cars, including most performance sedans.
In cold weather, which is again from late fall to early spring the new winter tires offer very good wear and are as quiet as your summer tires.
Finally, All Season tires are not considered to be winter tires and there are road checks in bad weather conditions where the police will turn back cars without winter tires.
If you are interested in reading more about winter tire advantaged, here is a good summary and FAQs. Although more emphasis could be put on their advantages in cold wet conditions, their comments on the extra feeling of preparedness and safety are well thought.