Cats do not have conceptual consciousness; they don't have language and do not think in universals or concepts. By human standards their minds are terribly limited -- and yet, as limited as their minds are, no cat is exactly the same as other cats. Each cat has a style of doing things and little quirks of behavior that make him unique.
Cats form their personality early and then never change. For this reason, it's good to be careful when selecting a kitten. When you visit a litter, if a kitten comes to you, that is the one you want. It is friendly, unafraid and outgoing -- and will remain so until its death. The kitten that cowers in the corner and hisses when you approach will always be that way; you might establish some understanding with it, but it will never be friendly and outgoing.
A screenwriting teacher of mine at UCLA used to say, "Your hero can kick a cat, but not a dog." We recoil from abusing dogs, but we think abusing cats is funny. Why is this? In large part I think it comes down to the noise each animal makes when it is hurt. Dogs make a human-sounding YELP! and then they whine or moan. We sympathize with the dog's suffering. But as anyone who has stepped on a cat knows, they make an inhuman Satanic screech. That screech is just funny, especially on film. It's hilarious when the cat chews the Christmas lights cord in National Lampoon's Christmas Vacation.
That teacher also said that having a hero pet a dog is one of the quickest ways to establish that he is a good guy. In a movie that one image of petting a dog is worth more than a thousand words from other characters testifying that a character is nice.
Much of our enjoyment of pets comes from projecting our humanity onto them. We think they love us the way we love them. I usually call this the "pathetic fallacy," but looking the concept up on the web, I see that it comes from a confusing essay by John Ruskin about art. We'll set the concept of the pathetic fallacy aside for now.
The behaviors we think of in human terms come from one major difference in cats and dogs: cats are territorial, whereas dogs are pack animals.
As pack animals that roam across large distances without regard to territory, it is vitally important that dogs maintain contact with the pack. This is why a dog will run to his master when he whistles. The dog sees humans as leaders of the pack and he wants to make them happy in order to fit into the pack. The pack has a pecking order that the dog respects.
As territorial beasts, cats do not have the need to run to the master when called. Some cats will run to you if they are trained to think they will get food for it, but many will ignore a call or respond with a meow, as if to say, "I'm here in our territory, as I should be." My cats do not like it when I approach them; they prefer me to be stationary so that they can approach me on their terms and when they want.
The old joke, "Dogs have a master, cats have staff," reflects the pack/territorial distinction. Dogs need a leader of the pack to whose dominance they submit. Cats can be more aloof as long as they know you're there. Cats can also be more egalitarian, if you will, although some cats are certainly bossier than others.
As pack animals, dogs can go to new places without blinking. They will walk happily on leashes into any strange territory. To a cat, a new place can be traumatic. The cat hides beneath a bed or behind some appliance and can take days to explore a new territory.
Introducing a new cat into the territory of another cat can be traumatic and if the two cats get off to a bad start they might never be happy together. The way to do it is to put the new cat in a bedroom or bathroom with food, water and a litter box, then close the door. Let the other cats in the house smell the new cat under the crack in the door for a few days. Then leave the door open a crack and let the cats explore one another on their own terms. The worst thing you can do is throw a new cat into the midst of other cats, because then you get hissing and spitting and fighting. Dogs, of course, can become friends in minutes. Compared to cats, they're like, "Dude, let's party! Whoo-hoo!"
When cats rub your legs, even that is a manifestation of territoriality. Cats have glands in their cheeks that they rub on your legs to mark you as theirs. And you thought they were just being affectionate. If you'll notice, they also do it to the legs of coffee tables and other furniture. (Now, dogs humping your leg is a mystery to me and I'm not sure I want to know why they do that.)
As rational animals, humans think in concepts. We form values that we act to gain and keep. Do dumb animals have values? I think on some low, limited level you could say they do. My cats follow me from room to room because they know I'm their meal ticket. They like to crawl all over me and curl up on my lap. They purr when I pet them, which is a communication of affection. But what is petting to a cat? They think I'm grooming the parts of their head that are hard for them to reach.
I suspect that we project values onto our pets far more than they actually have. Cats and dogs are incapable of loving the way humans love. They can know fear, but not envy or hatred. If they know love or anger, it is within their very limited, perceptual context of knowledge. The rest of their behavior comes from their nature as a pack or territorial animal.
UPDATE: Slight revision.