Cats are thought to be self-domesticated; dogs, domesticated by us. A supposed test to measure your dog's intelligence is to put a towel on its head and see how long it takes the dog to remove it. A smart cat doesn't even let you get near enough to drape the towel over its head. An even smarter cat also gives you a nice scratch to warn you not to try to pull that kind of crap with it again.
Cats can however be trained to do impressive "tricks" just like dogs can. See the numerous youtube videos featuring cats that actually use the (human, porcelain) toilet, and then flush it afterwards. I kid you not.
Personally, I like to think cats are smarter. Their human-like behaviour just seems more mature than a dog's human-like behaviour. A cat is like a young adult, a dog is more like a toddler. Most dogs, if you let them get out the front door, they go barreling down the street and get themselves lost, or run out into traffic and get hit. Cats don't do this.
Also, this might speak more to who has more instinct, or perhaps just better personal hygiene, but it's interesting nonetheless: you don't need to train cats to use the kitty litter box, they do it instinctively.