According to lore, cats were hunted to extinction because they were an easy source of food. Not to mention, that the vermin that they used to prey on, mutated to become large enough to eat the cats. Radroaches, Molerats and Giant Rats would all kill and eat a cat no problem.
Large breed dogs survived the apocalypse because they were still useful to humans, and dogs form packs to protect themselves in the wild. Much safer chasing a cat for food, than to chase a dog and suddenly finding yourself running into the rest of it's pack (coyotes actually use this tactic).