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.
"We had a euthanasia drug similar to what is used for lethal injection and it was taken off of the market because it was cruel and inhumane."
Was it really cruel and inhumane, or was it just suggested that it was by annimal rights activists with no proof?
"I often wondered why we (veterinarians) are more caring about the animals than the Medical Doctors are about the prisoners about to be put to sleep."
All the evidence points to the current mixture of drugs working. The suggestion that people migh wake up is based on patients who received smaller than normal doses because of fears about their poor health. Convicts receive a much larger than normal dose, not a smaller dose.
Lethal injection does not take a long period of time like a surgery often does.
There are suggestions that other drug mixtures might work better. There is a long history of the current drugs working at their intended goal of killing in a humane way.
There's no reason to change what works.