Among many people there is a misconception that muzzle brakes and compensators are the same thing. This could hardly be further from the truth.
Firearm recoil is a combination of four different things;
1. Newtons 3rd law, When the bullet starts moving it forces the firearm to start moving in the opposite direction, this force only last milliseconds, and is proportional to Powder/Bullet Weight vs Firearm Weight.
2. Thrust, When the bullet leaves the muzzle the escaping gases provide thrust which allows the rearward motion of the firearm to continue. (The gas thrust is not sufficient, in most cases, to accelerate recoil, but it will allow it to maintain velocity already achieved)
3. Axial rotation, Except in rare instances, the barrel of a firearm is above the center of gravity of the firearm and any forces acting on the barrel are going to induce rotation around the center of gravity (commonly referred to as muzzle flip).
4. Levered rotation, Your shoulder is trying to resist rearward force, and if the barrel is above the point where the rifle contacts your shoulder, the Buttstock will “Lever” the muzzle/barrel upwards, as it converts linear energy into rotational energy.
A muzzle brake is designed to arrest the rearward momentum created and maintained by the first 2 forces and in turn prevent the 3rd and 4th altogether. The gases, upon bullet exit, are moving roughly 3-5 times the speed of the bullet and even though we think of these gases as not weighing anything, they in fact, do have weight/mass(roughly equal to the weight of the powder charge), and since energy is mass(1/2) x velocity squared, the weight/mass of the gases impact the brake with enough force/energy to stop the rearward momentum almost entirely.
A Compensator, Such as the ones you see on AR style rifles, competition pistols, and on “Brakes” that have top ports or biased porting are creating a thrust to mitigate rotation or flip. They accomplish this by redirecting the gases to create thrust. Thrust can be “redirected” by a compensator and, due to the leverage of the barrel length, push the muzzle in a given(opposed direction). Even though Compensators may reduce recoil, their purpose is to control flip or muzzle rise.
In finishing, A Compensator is like a steering wheel, it “steers” the recoil. A Muzzle Brake is like, well, a brake, It Stops the recoil, and IF it’s a Good Brake, Doesn’t Need to Steer to Keep the Muzzle down.