What's the difference? I won't dig deep, but we all know through moral intuition that killing a person is never the same as killing a dog. People are people, and dogs are dogs.
Dogs and other animals ought to be treated humanely and compassionately, but this does not preclude intentional killing (even animal rights activists agree with animal euthanasia). Human beings, on the other hand, by virtue of the sort of entity they are, have a profound and inherent dignity such that it is prima facie gravely wrong to kill them. It degrades that dignity to treat a human being as one would treat an animal at the end of life. It degrades that dignity to value human beings for what they can do rather than who and what they are.
That's why stories like this one -- in which a woman recounts helping her friend kill himself -- should make us sick. That's why suicide as a "solution" to life's troubles is inescapably sad and tragic, never an act to celebrate. Human life is different. Human beings matter more.
(* I define euthanasia here as the intentional killing of a patient by act or omission for his or her alleged benefit. This does not include, for example, allowing a natural death by withholding artificial treatment from a terminally ill patient.)