Canine and Feline castration involves the removal of the testes. It involves a day procedure whereby your pet will be under general anaesthesia for about 10 minutes in cats, and up to 20 minutes in a dog. With rest and pain relief for a couple of days, your pet is usually back to their normal selves very quickly.
Without testosterone, you male cat will not feel the need to urine mark / spray its territory. Your male dog will be easier to train, have a smaller prostate gland and have no testicular issues.
Vasectomies are currently very rarely performed, and most vets will only do them under specific circumstances. Similar to humans, it involves removing a section of the spermatic cord, thus preventing sperm from travelling.
After a vasectomy the testes continue to produce testosterone. This means that your pet will still have his inherent male behaviour. The most noticeable behaviours in dogs are seeking out females and aggression. In cats, it is urine spraying with a distinctive tom cat odour. Testosterone will also cause the prostate gland to continually grow, almost always causing problems in older undesexed / entire dogs.
In the wild, inherent male behaviour was necessary for survival of the fittest. In modern society, male dogs often destroy property whilst seeking out females. They will be a risk to themselves and other people as they negotiate roads. Male dominance aggression will increase the danger to all involved in these situations.
Vasectomised and entire male dogs will nearly all develop prostatic disease in old age. Diseases vary from benign prostatic hyperplasia to cancer, which is usually malignant and often already spread upon diagnoses.
Signs of any prostatic disease are similar despite the cause:
Diagnosis is based on ultrasound, x-rays, blood tests, and prostatic sampling. Treatment often involves admittance to hospital, long courses of medication (due to prostatic resistance to drugs) and castration is usually recommended, since removing the source of testosterone will cause the prostate to shrink. Chemical castration is a possibility however it is less reliable than surgical.
And so it is for medical and societal reasons that castration is recommended over vasectomizing dogs and cats. Castration is a relatively easy procedure to perform, fast and inexpensive, and in the long run has many benefits over vasectomizing pet dogs.
References: Update on Canine Prostatic Disease, Central Veterinary Conference 2013 - Washington DC, India F. Lane, DVM, MS, EdD, DACVIM, The University of Tennessee, Knoxville, TN, USA