The military calls the A2's birdcage flash suppressor a 'compensator' because it also helps reduce muzzle rise. The BATF says its a flash supressor so its a no-no on post ban rifles.
Muzzle Brakes are devices that are used to reduce RECOIL in a rifle. They also have a tendancy to increase the muzzle blast (particularly to the sides) and flash.
Compensators are devices that reduce muzzle climb. They help keep the rifle on target during a rapid fire string (or in the M16s case on burst or full-auto).
Now there are combination devices for example most post ban 'Muzzle Brakes' are actually a combination Muzzle Brake & Compensator. And as noted above the A2 Bird Cage is a combination Flash Supressor & Compensator.
Note for Muzzle Brakes or Compensators to be post ban legal they must not reduce the flash nor can they be 22mm +/-1 in diameter (to prevent usage as a grenade launcher).