An explanation that I heard as a graduate student was that the skull of a dog tends to undergo much greater change in shape than the skull of a cat as the animal matures. In particular, the snout tends to be short in a newborn puppy and proportionally much longer in an adult dog. Growth-related effects like this are often subject to a lot of evolutionary variation; a dog in which snout elongation starts early in life, proceeds quickly and ends late will end up with a much longer snout than a dog in which elongation starts later, happens more slowly and/or ends earlier, so there are a lot of variables in play.
In other words, the fact that skull shape changes during maturation leads to variation that breeders can exploit by artificially selecting long-snouted or short-snouted individuals. With cats, this type of variation in skull shape isn't available.
I don't know whether other aspects of body shape, such as the lengths of the limbs relative to the body, also change more during growth in dogs than in cats. However, the point about skull elongation is probably a partial explanation for the difference in breed diversity mentioned in the question.