So, differences that I've seen or been told about so far:
The armor mechanics are different: This is probably the biggest change in actual in-game gameplay, since many tanks are going to be outfitted with Reactive Armor this changes how you deal with them and what ammo types are going to be effective beyond WoT's angling mechanics. According to Obsidian it's also going to mean that shooting the same spot, say in a close range engagement, will result in less effective armor thickness at that location.
The ammo types interact in a more complex way because of the new armor types. WoT, in practice, basically has AP, AP++ (APCR), HE, and HEAT with the only differences being shell velocity and you don't fire HE or HEAT at spaced armor. From the look of things AW is going to have more types of ammo and more nuanced interaction between ammo and armor.
The game is made up entirely of post WW2 vehicles, which means that the absolute slowest thing we're likely to see is going to be in the 40+ KPH range. This doesn't sound significant but I think it's going to massively effect how the game plays out in practice, both in terms of the pacing of multiplayer matches and in the tactics people develop because even the MBTs are going to be able to re-position and flank to a very high degree.
Their damage system is different. By this I mean the amount of RNG in the damage is lower (~12% range vs WoT's 25% range) and they have a non-critical hits system for stuff like cupola sniping so those shots do less damage since there's nothing really vital about blowing off the commander's cupola.
Their crew and progression systems are completely different. This is probably the biggest change overall. They're basically replacing WoT's crew system with a more RPG like Commanders system and the progression is going to be completely different both for crew and for acquiring new tanks.
Arty is going to play completely differently. They're currently messing with player feedback from their first focus test to try and develop a more interesting and less frustrating arty system than WoT's. I think this combined with the more mobile battlefield is going to give AW a very different feel from WoT.
Extensive PvE content promised on launch. AW isn't just a PvP game like WoT is and this, more than almost anything else, is probably going to attract a very different player-base from WoT as well as introduce a whole host of community conflicts and issues. Think "how did this tomato get a T29!?!" but on steroids because you won't necessarily have to play PvP at all to get high end vehicles.
So yeah, that's basically everything I could think of in 15 minutes. I'm sure other people have other stuff to add or their own perspectives on the differences I've pointed out. Personally though I think it adds up to a pretty different game.