Quiet: You make a sound but not a loud sound.

Example:

✗ I have a car with silent engine. (silent engine = broken car)

 ✓ I have a car with quiet engine. (all cars make sound, however this car doesn't make a loud sound -> quiet)

Silent: You don't make any sound at all (no noise, no sound)

Example:

 ✗ She didn't say anything, she was quiet.

 ✓ She didn't say anything, she was silent.