To put it my way, catapults throw, trebuchets fling.
A catapult launches its projectile somewhat like throwing something that is cupped in your hand with a straight arm. A trebuchet is more like holding a string attached to an object, and swinging your arm up and letting go of the string. Also a trebuchet uses a large weight to move the arm, the catapult is more like having a spring.
This was probably kind of confusing, but I hope I was able to help :)