For one thing a regular miter saw, like those cheesy box style miter saws, can cut on predefined angles only. Over time, this angle is no longer as accurate as it was originally. Furthermore, the saw loosely fits in the slot so there is play so the angle is not precise.
On a compound miter saw you do not have to cut on predefined angles only - you have every angle from 0 degrees to 180 degree available to you. Additionally, the saw can tilt to make a straight cut across the board, but with a 45 degree tilt, for example.
So yes, you can turn the saw to any angle on the circle and tilt it to any angle upto a 45 degree tilt I would think ( I may be wrong on how far it tilts).
So compound means not just one angle can be cut but two angles at the same time Example, I can cut molding at a 45 degree angle with a 45 degree tilt - all in one cut.
It is furthermore a very precise cut since you are locked on the angle and the blade cannot move, no filling in gaps afterwards.
I would always choose the compound saw over the conventional miter saw. Even setting up a jig for a saber saw is a better cut than a conventional box style miter saw.