I heard a saying (by someone famous - George bernard Shaw???) that sums up some of the differences between cats, dogs...and pigs!
"Dogs look up to you, cats look down on you, but pigs treat you as equals!"
I'm not sure if this is a measure of intelligence or not, but cats, unlike dogs, are not pack animals. Cats are too smart to do tricks just because we want them to! They don't have the urge to please their pack leader as dogs do.
Pet cats often seem to look on the people of their household as dull, but occasionally useful, friends. Like Sherlock Holmes with Dr Watson. We do provide food, a warm home, toys, good places to scratch and a comfortable lap (or bed, or whatever) when required. But we are absolutely hopeless at catching mice; in spite of all their efforts to teach us. Even a kitten learns that art better than a human, and practices diligently at every opportunity!