Being a tire dealer I`m going to try to stay impartial here.
There are distinct differences. The All Weather tire is really a hybrid of All Season and Winter.
All Season tires have a relatively straight uniform tread pattern designed to provide a good ``all `round`` performance in dry and wet weather. The tread pattern also helps to keep down road noise. These tires have the hardest compounds and longest life.
All Weather tires are relatively new to the market, perhaps 5 years. They have a much `blockier`tread pattern, and softer compound. The blockier tread and softer compound provides better steering and stopping performance in wet weather (and slush) however the compromise is they are noisier and have shorter life.
Winter tires, I think most are familiar with. They have in general a uniform block tread with a much softer compound. They offer significant improvement in grip, steering and stopping distance in ice and snow however they are noisier and have much shorter life. If you are buying Winter tires spend a few hundred dollars more and get a set of dedicated rims. Your tires will last longer not being taken on and off twice a year but the changeover is significantly less expensive. Just a note on Winter tires. It is not the air temps that kill them it`s the temp of the road so get them on when the road temps have had a chance to cool down and more important get èm off before the roads heat up! I have had customers drive to Florida on Winters stay for a few months and complain when they get back that their tires are wearing too fast...