There was a BBC early-morning breakfast tv slot in which a dog was presented as 'vegetarian', until test time...the dog (given free choice between two bowls of food placed on the studio floor) went immmediiately for the meat dish.
From my understanding, it's generally for health purposes (though the convenience doesn't hurt). An example my vet provided is that the level of sodium consumption needed by humans is way too high for dogs.
There's an economic benefit: in the UK (and many other countries), ingredients which are not "fit for human consumption" (and might otherwise be thrown away) can be processed into pet food.
Much commercial dog-food is made with ingredients which aren't fit to be consumed by humans.