They don't necessarily do the same thing.
A flash suppressor will just help mitigate the muzzle flash, but not necessarily reduce muzzle climb as a muzzle brake would. Basically, if the bottom of whatever is attached to the muzzle is open (via slots/slits/holes/etc.), meaning gases can go down, then it's likely not a muzzle brake, as gases jetting downwards would do more to add to muzzle climb rather than help eliminate it. If the thingy at the end of the muzzle is closed at the bottom, it's a muzzle brake, since expelled gases would tend to push down, somewhat countering the muzzle climb.
One example would be comparing the flash suppressor on the new Ruger SR 556 with an A2 flash hider on another AR-15 design rifle. The Ruger SR 556 basically uses the same flash suppressor that's available on their Minis, which is a suppressor only, and doesn't really act as a muzzle brake since the bottom is not closed. The A2 flash hider acts as a muzzle brake as well, sine the bottom is closed (no slots/slits).
Or something like that.