Question: My cat catches a lot of mice and birds. It always leaves the heart, perfectly intact. My question is, what about the tails of the mice, the feathers of the birds? I sometimes find some feathers but never all of them. Does the cat eat it, and why does it leave the heart?

Answer: 
Cats are very picky with what they eat. Although hearts have a lot of good nutrients for cats, if a cat doesnt like the texture or taste, it might just reject it. Cats most likely won't eat the feathers or the tails, as they are harder to digest, but might occasionally eat a few. 

Although this is unrelated, and not a current issue for you, it will prevent future stress. Remember to get your cat regularily checked up on by the vet. Mice and birds can carry diseases.
Maybe the hearts are too tough to eat even with their sharp teeth, or maybe they are actually poisonous and therefore inedible. If either is the case, the cat knows, and therefore leaves it alone. As for the feathers, I think they are discarded since they're most likely as useless to the cat as the hearts.