I think the reasoning behind it is the same (to end pain), but the criteria for implementing it is different. Obviously one of the major differences is choice, and humans can choose for themselves whether they want it or not. Considering those differences, euthanasia with a human is a much more complex decision. The ability to give consent is not always black and white (mental capacity), and of course there are things like living wills, and disagreements between family members. But overall, it’s the same, I believe, and I think people should have the option for euthanasia if they have specified it in a living will or have been determined to be of the mental capacity to make such a big decision.
Having said that, in both cases, it should be done in a humane manner that is comfortable for everyone involved, especially the person/animal receiving it. I do not believe animals to be lesser beings than humans, and as such, they should not be treated different.