Some animals have fins and some animals have flippers. Fish have fins. Mammals like seals or dolphins have flippers. A penguins "wings" are also called flippers.
If you are talking about a long, linear extrusion from a mechanical device such as a car or a missile, then those are called fins.
But the movable mechanical object such as the part of the pinball machine are flippers.
So summing it up, here is what it seems like to me:
Fish have fins. Other water animals have flippers.
Non-animals are harder to explain, but here, I only think of something as a flipper if it is used to flip something (like a ball).
(Thinking again on dolphins: They have both flippers and a fin. The tall fin in the dorsal area is called a fin, but their tail flukes and the parts on the side or underneath their bodies are called flippers)