You STAY there until I come back to you

The dog stays in one place without moving until you return back to the dog and release it.

The dog never moves.

You WAIT there until I tell you what to do next

The dog waits in one place until you tell the dog what to do next.

The dog will always move.

STAY is a formal exercise and the body signal to the dog is  standing with your side or back to the dog.

WAIT is a less formal exercise and the body signal is to stand face on to the dog.

Examples of when to use WAIT

Wait while I put your lead on

Wait while we cross the road

Wait while you open the door

Wait while I put your food down

Wait until I throw your toy