If a car overheats and cuts off is the motor completely damaged?

No but you will need to buy and replace the thermostat. Shouldn't cost more than $5.00 with tax.

Not always the thermostat, it could be vaporlock. You could have air coming into your cooling system do to a hole in the radiator or a leak in a hose. But no serious damage unless you keep running the car hot, you could blow your head gasket, and that can get messy, and rather expensive.

ANSWER

Both of these are completely wrong. I am an ASE master certified technician with an automotive degree. I currently run a repair facility. I have never heard of vapor lock on a coolin system. It is impossible. The thermostat could have caused your initial overheating, but would not cause it to die. 98% of the time your engine has been damaged. If it restarts and runs rough, hopefully it's just a head gasket. If it will not start at all, more than likely you warped your head beyond repair. Alot depends on the vehicle and if it is an aluminum head or cast iron. Neither of these individuals should be answering automotive questions.