They're really different games. AB is a real-time, lower-level tactical game - it's probably most similar to Close Combat type games, except played on maps that can get pretty massive (15.25x15.25km) in size, and does not rely on much of anything pre-scripted - whereas Flashpoint Campaigns goes into a bit more depth with setting up its scenarios by design, AB's approach is more "your force is on this side, their force is on that side, here's some objectives, go!". It's a much more "what you see is what you get" kind of game, as opposed to Flashpoint Campaigns which requires you to do a bit more interpreting of the info you're getting on your own and enemy units.
The nice thing about AB is that it's pretty versatile and sandbox-y, in the sense that you can throw pretty much any kind of units at pretty much any kind of map, and it'll do an adequate job of fighting that battle with minimal effort to set it all up (just a couple of minutes picking options in the scenario or campaign generator). And if you've played games like Close Combat, Combat Mission, and the likes - the controls and way of playing the game will feel pretty familiar. The scale and pace of AB is totally different from those, but how you control units and get things done feels very similar, so it's not hard to get into.
Otherwise, Flashpoint Campaigns is a more complex, more hands-off kind of "command simulation" game with a complex asynchronous turn system, whereas AB is more of a hands-on real-time tactics game.