The short answer is: there is none.
It’s all nomenclature; the stuff that covers the bodies of many of our pets and their wild cousins is made up of the same stuff that covers parts of our bodies: an outgrowth of the protein keratin. We just typically call it hair when referring to it on a human body, and fur when referring to non-human mammals.
For some reason I tend to think of my cat’s hair as fur while it’s still on his body, but hair once it has become embedded in every square inch of my bedding, couch, and carpets. And I’m more inclined to think of dogs, cats, ungulates (hooved animals), and apes as having hair, whereas I always think of bunnies, bears, foxes, wolves, and beavers as having “fur” (yes, I know – foxes and wolves are dogs…it makes no sense).
Some fun facts about animal hair:
- Porcupine quills are greatly enlarged hairs.
- Whiskers are hairs that work as sensory receptors.
- Fingernails and claws, rhino horns, and the scales covering the pangolin are all made of the same stuff that hair/fur is made of: keratin.
- Not all languages differentiate between hair and fur like we do in English.
Check out this Mental Floss article for more interesting facts about hair/fur.