Now that I'm not on my phone: The biggest difference as has been mentioned above, will come in hardpack and ice. ATs have enough tread to dig through softer snow and when paired with chains can perform adequately, but the silica content is too low, so the rubber doesn't maintain enough pliability at low temperatures. Basically below 7C they turn into hockey pucks and don't grip anymore. Winter tires are rated to much lower temperatures, in addition to the changes they make in tread patterns like increased siping and provisions for studs.
As far as winter tires, depending on the load you'll be carrying, my recommendations are as follows:
Bridgestone Winterforce W80 (I think that's what they're up to, may be W90 now)
Hankook iPike RW11
Hercules Avalanche Extreme (honestly, the best bang for the buck winter tire I've come across, and I've tried many)
BFG Winter Slalom
I know the first 3 are available in LT sizes for load carrying capacities required on trucks, and IIRC the BFG is as well.
I have extensive experience on all of those tires and I would say the Hankook is the best of the bunch, but you pay dearly for it. Avalanche Extremes are the tire of choice in my area as they're fairly inexpensive, wear amazingly (I know people who run these year round on fullsized trucks with great success), and they have good traction in both the deep snow and on ice, which we see lots of.
In areas with more hardpack/ice I would shift my recommendation and include the following for a non LT application:
Michelin x-Ice
Nokkan Hakkapelita (probably spelled it wrong)
Both of those are tops for hardpack/ice, but I don't believe they come in load rated applications, and they don't perform as strongly as the previous suggestions in deeper snow.
I am a tire geek, and career mechanic, I also spent the early part of my career building tires in a retreading plant. It's kinda my thing and those are my suggestions.
Edit: All of the tires I suggested are capable of being studded with the exception of the Nokkans, and all perform adequately without studs, but if you see a lot of ice, I would recommend studding if it's allowed in your area.