There's not a huge amount to say, except:
1. AoD calculates an effective supply efficiency for each province in real time.
2. AoD uses an additional stacking penalty to simulate the amount of force an attack can deliver. As a general grand-scale military strategy rule of thumb, the weight of an attack increases with the square root of its size; by adding stats behind the scenes that determine how concentrated a division's force can be (armor being the clear winner on that front), AoD makes mass infantry attacks a lot more deadly for the attacker than other HoI2 games.
I'm pretty sure Darkest Hour does recruiting differently, too; raising a division takes quite a while unless the country is mobilized, which is a mechanic entirely absent in AoD.
HoI3 is a different beast than HoI2 games entirely--the map is divided more finely (10,000 individual provinces, to Darkest Hour's ~2700 [?]), for one. I haven't gotten around to playing it yet, since I'm planning on waiting for a compilation edition, and my games backlog is already full to bursting as it is. It does have three features I like, though:
1. Build your own divisions. You recruit brigade-by-brigade and combine them into divisions, which is really cool.
2. Chain-of-command. You set up your own order of battle with commanders at various points along the tree (so I understand).
3. Aircraft carriers. In HoI2 games, they're very abstract--they're basically battleships with a very long range, a very high Air Defense value (for shooting down enemy airplanes), and a few special missions for attacking shore facilities. In HoI3, you take real air units and attach them to the carrier; you can then use the aircraft just like land-based ones.