A workout is usually in a gym with exercise machines, or at least you follow a regular routine of exercise. Exercise can be anything: from going for a walk, to playing a sport, or going to the gym, etc.

A workout is exercise. But exercise is not necessarily a workout. When you go to the gym for example, you go there to exercise or to workout (in this case they both mean the same thing). If you go for a walk, hiking, play a sport, then you would say that you are exercising and NOT working out. When people use "working out" they usually refer to exercise that is part of a training routine. You can in most cases substitute the word workout, with exercise, but not the other way around. "I worked out this morning" or "I exercised this morning" are both fine and mean the same thing.