Dog Breeds Outnumber Cat Breeds
Photo by mickeymox - cropped as allowed under creative commons license
Dog breeds outnumber cat breeds for these reasons. Firstly, dogs have historically been used to work on our behalf. Dogs are more trainable than cats and generally stronger. This makes them more suitable as working animals. Which would naturally lead over time to breeding specialist dogs to do certain types of work. The classic role for a working dog is to support a person who is hunting. Examples are pointers and hounds. In addition dogs are used to pull loads, clear rats, seek out drugs, help fishermen, help the blind, guarding people, assist in hearing, help in psychological therapy (other animals can assist in psychological therapy as well) etc..
From that early foundation of breeding different working dogs, the "dog fancy" as I shall call it (i.e. dog lovers and breeders) bred all manner of dog breeds simply for the satisfaction of it and their interest in the dog. This would clearly have greatly expanded the number of breeds.
However, unregulated or irresponsible breeding can and has led to genetically inherited health problems. The recent (late 2008) BBC television programme exposing genetic defects in a number of dog breeds comes to mind. Similar problems, but on a lesser scale it seems, have occurred in the cat world (Modern Siamese and Persians, being examples).
Having expanded the number of dog breeds substantially dog breeders would then have more opportunities to create hybrids. Hybrids are nearly always man made breeds by crossing one breed with another. The growth in breeds can accelerate as more breeds are created. There may have been a desire to try and eliminate defective recessive genes causing health problems by mating healthy breeds with less healthy ones, thereby creating more dog breeds.
Whereas there are between 170-400 dog breeds, depending on the registration of the breeds, there are only something in the order of 40-100 cat breeds and most cat breeds beyond the mainstream 40 are very rare indeed and really experimental. So dog breeds outnumber cat breeds substantially.
Of course, the cat fancy follows the same desire to create new breeds but it may be that the cat associations better regulate the creation of new breeds by hybridization thus restricting the growth of breeds. But it seems to me that the underlying reason why there are more dog breeds than cat breeds is because, early on and thousands of years ago, the dog was found to be more suited to support people in their work and play.
However, the cat's domestication came about because of their skill as hunters thereby assisting farmers in keeping down rodents where grain was stored. Cats can work too but this is instinctive and not a trained activity.