Does neutering a dog make a difference?

Well the main thing is that the dog/cat won't be able to reproduce. I've heard many things about neutering your pet. The one that interests me the most was that supposedly it will reduce the chance of your dog/cat running away. Dogs often run away because they are looking for a "bitch" (female dog) to mate with. Therefore neutering him will make him not want to find a mating partner. (To simplify it; the dog wouldn't be horny and wouldn't go look for a female dog to have sex with.)