The person in coma is alive and can wake up. The brain is still perfused. And that's the big difference to brain death.Due to no brain perfusion, all the brainfunctions of a braindead person are completely and irreversibly extincted. There is no external difference.
Because of medicamentous support of all the important bodyfunctions, the circulation and respiration can be sustained for some hours. Without any of this ICU-support, apnoea would occur immediately after brain death, followed by cardiac arrest.