Because the animals that you put down are innocent. Their only "guilt" is to have contracted a painful condition, a disease, or lived too long to be able make it to the back yard. Furthermore, the beast that you dispatch is much more than the sum of its biology to its owner. The circumstance of a pets death will linger for a long time in the mind of its owner and you do your business wise by being just as respectful of the spiritual bond that you are breaking as you are of the physical.
As for the criminals, they don't deserve a "good death", but society does so that executions can be carried out without cause of interference from outside groups. Guilt should be without doubt and so should the absence of cruel and unusual.
Ahh....the whole article is specious at best.
It is predicated on the person 'waking up' from a too small dose of Thiopental before he's dead.
Give him more thiopental, and the problem is taken care of...not to mention that within 45 seconds of cardiac arrest the lack of blood flow to the brain will render him amnesic anyway.
As someone who has administered a LOT of thiopental, I can tell you that even with a commonly accepted anesthetic induction dose of about 4-5 mg/kg (and I would bet they give even more) that the person would not be remembering anything for at least 15 minutes (in all likelihood longer) and in that time the person would be dead from the potassium and induced cardiac arrest.