Dogs are also predators. Quite a lot of large dog breeds should not be kept with cats, because they have enough prey drive that the cat will be in very grave danger. Dogs were historically bred for ratting and gophering.
I'm thinking the reason dogs don't go for birds is that the birds are too small and fast; dogs are predators, but they aren't ambush predators like cats.
Dogs are also pack animals (as wolves) and so they can be trained easier by abusing the pack instincts, even unintentionally.
Cats are solitary and so even if you get them to not do something in front of you, once you're out of sight they'll quickly revert to doing whatever it is they want to do.
I'm thinking the reason dogs don't go for birds is that the birds are too small and fast; dogs are predators, but they aren't ambush predators like cats.