I don't have TOAW4, although I have each of the three previous versions. There are a couple differences.
1. TOAW is basically an operational level game with a strength on specific historical scenarios. By operational, I mean a planned action in a specific geographic area during a fixed time period. The objectives are defined by the scenario rule, i.e., your superiors. Your role is to assume command of the operation. You don't have that much control over the makeup of your forces, or future reinforcements. There are upper-level commanders making these decisions for you, and you have to execute the operation. ATG can do this as well, but through user-made scenarios, and not out of the box, so to speak. This is basically the core of TOAW. It comes with many scenarios.
2. ATG can be extended to a strategic game with its economic engine where the player determines production and technological research. This is also done with random maps, giving a full war between regimes, and not just a specific operation. It is sort of a difference between recreating history or making an alternative history. In this regard, I think ATG has more replayability, as the random games are different every time. It isn't a matter of trying a different move, or hoping for a better random number this time; the whole world is different.
3. TOAW has a very extensive library of equipment, and I think it "scales" more easily as the hex and time scales change in the scenario. ATG can have an extensive library as well. But, it is something that the modding community will have to do, as it is not part of the core game.
4. What looks like to be a fairly big difference is that TOAW4 uses a movement then combat system. And movement is based on movement points. You move and plan a serious of attacks, and then if there is time left in your turn, you can move and fight again. In ATG, movement and combat take up action points. You can move several units, and then conduct attacks, or you can attack a unit in place and then move another unit.
5. From the TOAW4 tutorials, which look to be impossible to follow, it looks like TOAW4 might have a more realistic time-management system. Namely, that the movements are synced up to determine how time is used. For example, if a unit is moved to block the retreat of a unit, and it takes all of its movement to get there, then that attack would use up all the turn.
6. TOAW4 looks to have put effort into improving naval units and their combat. ATG AI doesn't use ships well. It won't use carriers, and it won't use surface ships to bombard a hex prior to an amphibious landing.
7. Being newer, I suspect the TOAW4 AI is a bit stronger. But, it generally has less to worry about, namely, it doesn't deal with production, and the long-term strategic aspects. And, the ATG AI has improved its combat. The ATG AI also has some issues in structure of the units. For example, it will sometimes fail to optimize a mixed unit's movement capability.
8. In many ways, the Decisive Campaigns series is a set of operational campaigns, much like TOAW, based on the ATG engines.
9. ATG is moddable in many ways.
Like all things, it depends on what you are looking for. If you want to play many different detailed historical actions, then TOAW4 is a good choice. If you want the random game, and lead your "country", then ATG offers something few others can. Same if you want to mod the heck out of it. ATG, though, can also do the scenarios, it just needs a lot of work put into it do so.