When things happen separately, they occur on an individual basis. Students who study separately study on their own. If you buy a remote-controlled helicopter and the box says “batteries sold separately,” you’ll be land-bound until you get some AAs.
To separate things is to take them apart, like separating two students by making them sit in different parts of the room. Anything that occurs separately also exists on its own. If the President meets with members of Congress separately, he meets with them one at a time. If you don’t like it when your food touches, then you should eat the mashed potatoes and salad separately, not on the same plate.