Why is it humane to put down a lame horse, or sick pet, but for humans, it's a crime?

[Moonbeam - butterfly]

Why as a society, we find it perfectly okay, and in fact claim it to be the humane thing to put down a race horse that breaks a leg, or a beloved family pet who is suffering from pain and will never get better, yet when someone we love is in agonizing pain, and will never be better, it's a crime to allow them to die with their way with dignity intact, we, keep them alive , for what? Aren't we a cruel & backasswards society to allow this?