The question isn't so much, which one is better, as how do they differ. They are very different games, trying to achieve different things, and each one pretty much succeeds at what it is trying to do.
FOF is strategic with the option of tactical battles. AACW is strategic-operational.
This results in very different maps. The FOF map has the major strategic zones, e.g., Middle Tennessee, Shenandoah, key stretches of river such, etc. AACW subdivides the same zones into "counties" that allow for operational maneuver. Some say that the choice between the games depends on whether you want tactical battles, but I totally enjoy FOF without ever having played the tactical battles. Others say it is the icing on the cake.
Both games have micromanagement, but of different aspects of the game. In FOF, you manage the economy, diplomacy, even the selection of weapons for your troops. In AACW, there is a somewhat complex army organization; there are many actions that are perhaps sub-strategic such as destroying and repairing railroads, and returning your ships to port fairly frequently for repair and supply.
In FOF, there are many options, so you can let the AI do almost anything, as well as fine tune certain aspects of game play, such as how long sieges take. AACW has a few options, such as FOW at different settings, one of which reduces the micromanagement of the naval aspects. Even at full micromanagement, I find I can play a turn of FOF in about half the time of AACW, since the latter has more units to attend to than the latter.
One of the major differences is that AACW is much more historically scripted, with leaders and armies arriving when and where and in the precise command they did historically, and events somewhat driving the action. FOF is more open-ended, which I happen to like, but many like more scripted games that have a bit of the flavor of historical reenactment. For example, in FOF, you place any available general (with sufficient stars) in any army, corps, or division. In AACW, they appear as they did historically, though you can subsequently reassign them.
In my experience, both games have really excellent AIs.
They are very different experiences, and both illuminate key aspects of the American Civil War. If you can, get both!