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.
Cats remove the feathers by licking them, and they’ll inevitably swallow some of them in the process; their mouth structure doesn’t really allow spitting stuff out like ours does.

If it’s “your” cat, I assume you feed them? If so, they hunt primarily because their instincts tell them to, not for nutrition. They may eat the tastiest/easiest parts of their prey, but they’ll leave the less tasty or more difficult parts. Feral or stray cats will usually eat the entire prey, including bones, because they *need* to.