It's not the end of the world if the cat chews down a few pieces of dog food, but it shouldn't be a steady diet, because cats have vastly different nutritional requirements than dogs.
Cats require more protein than dogs do.
Dogs are considered omnivores. This means that while they might prefer to eat meat, they can survive on plant material alone and they can digest carbohydrate foods. (That doesn't mean that a plant based food is a good source of nutrition for dogs. They can survive on it, while a cat cannot.)
Cats, however, are carnivores. They cannot sustain life unless they eat meat in some form, and they need a higher percentage of meat in their diet than dogs do.
Dogs have the ability to break down carotenoids (organic pigments from plants) and convert these into active Vitamin A.
Cats process little to no enzymes that break down plant-produced carotenoids. They have to get vitamin A that has already been converted from carotenoids to its active form by some other animal.
Arginine is a building block for proteins vital to many of an animal's internal functions. Dogs can produce enzymes that aid production of Arginine, and they do fine on a diet low in Arginine. thus, it's not a priority in dog food. Cats, on the other hand, cannot produce Arginine, and they are dependent on getting it with every meal.
Taurine is another amino acid, distributed through most body tissues. It is important for the heart, retina, and other organs. Dogs make their own Taurine, but cats must eat it preformed. It is not present in plant tissues, they can only get it from meat.
These are just three examples of substances cats must have that they won't get in sufficient amounts from eating dog food. Other examples include arachidonic acid, niacin, and felinine (a compound made from a sulfur amino acid that is only present in cats).
~ Maria Sadowski ~