If an individual can process, identify, store information, and recognize differences, they are self-aware. Thats the definition among some psychologists and biologists.
Which most of the species listed in your link cannot do, not to any serious extent. Chimpanzees and dolphins yes, dogs and cats, no. This is of course a huge matter of debate and nothing is definitive (as is typical with biology and soft sciences like psychology).
Not that it matters anyway, I'm very much a "fuck other species" kind of person and could care less about animals.
Obviously they do not have the same level of consciousness so yes, they wouldn't have the same rights. However, a Neanderthal may just be as intelligent as us so how would you justify using her/him for scientific experiment. Though it would be extremely interesting to see what we can learn from her/him, it does feel a tiny bit wrong. Imagine being the only of your species, treated like a lab rat, be an outcast, and have very little freedom.
We experiment on people unwillingly all the time, especially here in the United States.