The main advantage to neutering the male dog is prostate health. The action of testosterone on the prostate of an un-neutered dog often causes enlargement over time (benign prostatic hyperplasia) and also predisposes the prostate to infection (prostatitis). Both of these are uncomfortable conditions that usually can only be resolved by neutering. Neutering also prevents cancer of the testicles and some anal tumours as well.
The neutered male dog is also much less likely to roam, be aggressive towards other dogs or urine mark. In these times dog control laws are much stricter and the emphasis is very much on the owner being in control of their dog at all times and in all circumstances.
Spaying the female dog before her first heat is also strongly recommended. Everytime an un-neutered bitch has a heat it greatly increases her chance of developing mammary cancer later in life. Also the action of the hormone progesterone on the uterus predisposes the mature unspeyed bitch towards developing the potentially life-threatening condition pyometra (womb infection).
Neutering the female cat is very important in order to control the number of kittens and therefore stray cats. Unspeyed female cats also have an increased chance of developing mammary cancer. They are also more at risk to contracting the Feline Aids virus.