What is the difference between Medically induced coma and medically paralyzed?

The brain is a funny thing. After an injury, the brain is under a lot of stress and can swell or react in ways that can cause further damage. Swelling in the brain causes more problems than say, if you twist your ankle. You're ankle can swell a lot without causing damage, but your brain is inside your skull, so if it swells, it doesn't have much space to go and pressure will build up inside.

Increased pressure in the brain can cut off blood flow to parts of the brain and can kill healthy brain tissue, which can lead to a secondary injury, which can be worse than the initial injury.

Doctors will give the patient sedatives, inducing coma to reduce swelling and pressure and to help minimize further damage to the brain. If the brain is "sleeping", it is resting and will reduce the need for blood and the overall blood flow to the brain.

Less overall blood flow to the brain + less need for blood = less swelling = less damage.

This is my understanding. Can definitely be improved upon. Hope this helps.