There’s really not much in it. The ratio of brain to body size is slightly higher for dogs than cats, which may be because dogs are a social species and have evolved some extra brain to handle more complex communication and interaction. But cats have almost twice as many neurons in their cerebral cortex, which is the area involved in memory, attention and perception.
Cats are generally harder to train to do tricks than dogs, and won’t fetch things or guard their owner. But does that mean that dogs are smarter, or cats are?