Both are descendant from apex predator animals that compete for food. Animals run away from larger animals, when a predator sees a smaller animal running away its their natural instinct to chase it down and use it for chew toy.
By in large it is how their social systems work. Dogs are very hierarchical where there is a dominate member and all others are submissive. Cats are - for a better word - cooperative. "Quid Pro Quo" is a better term. Cats work on the basis of if you do something for me, I will do something for you.