The main reason you should not feed cats and dogs the same food is that they have different nutritional requirements and pet food is formulated to meet these different requirements. This difference in requirements stems from their differences in metabolism and metabolic adaptations. Dogs are omnivores and cats are obligate carnivores.
The canine digestive tract has developed characteristics that allow digestion and usage of a diet with a variety of ingredients. For instance dogs have enzymes to digest starch effectively and so tolerate dietary carbohydrates well.
Cats on the other hand have dietary requirements that other mammals do not such as their requirement for taurine and arachidonic acid. Also cats have a much higher dietary protein requirement than dogs. As carnivores, cats must have part of their diet made up of meat to obtain nutrients such as arachidonic acid while dogs do not.
If you only feed your cat a pet food made for dogs, your cat’s nutrient requirements will not be met and your cat will suffer nutrient deficiencies and die.
-Jennifer D'Amato-Anderson, MS (Animal Nutrition)
Reference: Applied Veterinary Clinical Nutrition, First Ed. Edited by Andrea J. Fascetti and Sean J. Delaney.