In addition to Skippy's excellent information, one thing I've noticed is that there's a much higher likelihood of hormonal issues with later-desexed animals. If the cat or dog is mature when the operation happens, their hormonal balance tends to "freeze" at the point where they were spayed or neutered. 

I've seen this lead to compulsive over-eating, permanent "in heat" behavior, and other similar issues. 

The cats I've lived with that were fixed before maturity (usually before 4 months - the vets I went to would spay a cat from 12 months on, and neuter "as soon as the testicles dropped enough to see them" (this was my brother-in-law, who is currently a vet in the Brisbane, Australia area)) never had any of these issues.

Also, the cats I've lived with who were fixed young tended to be more affectionate and kittenish throughout their life where those fixed older tended to be less inclined to bond strongly with humans.