If you don't use one, you are taking a chance that your cat will leave without you. And most veterinary clinics insist that all dogs be on a leash. A dog who changes his mind coming through the door can back right out of his collar. Leashes prevent dog fights, whether your dog is the fighter or the fightee. Many dogs think house training doesn't apply in veterinary clinics.
Male dogs think veterinary clinics are a mecca for marking territory.
We can see this coming. He stands a little sideways, sniffing at the target, and then hikes his leg - on the most expensive sack of prescription dogfood in the place.