Dogs seem to hunt pretty much anything. I wonder if we've bred out the understanding of what is and isn't edible in dogs, but not cats? Or perhaps its a social thing that wild canids teach their young?
Half the dog owners I know have some story of their dog catching a hedgehog or something equally inedible, but the worst that seems to happen with cats is their prey fights back a bit harder than they expected and they get a rat scratch.
My experience with Labrador Retrievers is that they will eat anything, on the assumption that if it doesn't agree with them, they can always puke it up at 3AM.
Dogs were bred / created to keep their owners safe, not themselves. That seems to track. I've lived several places I'd not live without a dog alarm or three, so I can't say I'm not grateful.