Anything described as continuing is ongoing; it's doesn't stop. A continuing rain goes on and on and on.
When something continues, it doesn't end, so anything that is continuing lacks an ending. Comic books feature the continuing adventures of Batman, Spider-Man, and other heroes: the comics keep going without an endpoint. A chronic illness could be described as continuing: it stays with you and doesn't go away. If you have continuing money problem, your money problems are endless: you always have them. Most things end, but not continuing things.