I'm probably the odd man out here. While I agree that animals and humans have some qualities in common and I don't know what they are exactly, nor would I speak so confidently. People who speak too confidently, in my opinion, have too much faith.
With that said, there are marked differences. Let me give a few examples. One of the things that I have been studying for the past few months is the history of laughter and it occurred to me that this might be a distinct human quality. We find things funny and we tell jokes. Do animals? I am not sure. Also when we are confronted with something awesome and beyond us, like pictures of the universe, we are filled with wonder. Do animal wonder at the beauty of a sun set? Probably not.
One final example. I am also studying the difference between envy and covetousness in the ancient world. It is a fine distinction, but again do animal covet and envy? Probably not. The previous answers speak of continuity, I like to talk the about discontinuity. It is a matter of perspective.
The previous post did a very nice job in linking the animal experience to the human one. I would like to amplify this to a certain degree. Part of what allows humans to recognize characteristics of their own experience in that of animals might lie in a natural scope of empathy or compassion. This can be seen real well in films. A few years ago a film called, "March of the Penguins," became an extremely popular movie because humans were able to empathize with the experience and loss of children and search for companionship. In this setting, humans were able to see their own experience mirrored in that of the penguins. In the final analysis, such compassion and understanding allow us to understand that both animals and humans have shared characteristics, emotions, and experiences.
Your question is very 'us and them'. It suggests we are one group and animals are another. This is too simplistic.
You ask about 'human qualities' but you don't say what they are. I suppose you mean things such as intelligence, love, empathy, altruism etc etc
All animals (including humans) evoloved. Some animals are quite primitive, (eg bacteria, jellyfish, frogs, etc). Some are not (eg gorillas, whales, pigs etc) Some animals are closer relatives to humans than others.
Chimpanzees, our closest relative, are capable of many 'human' actions, and with successive generations of education can be taught to display a lot of emotions, feelings and thought processes.
There is no clear demarcation between human qualities and not so human qualities. It would be interesting to not that qualities and behavior described as inhuman is frequently found only in humans. For example, it is only humans who kill for pleasure.
Humans are also animals. Perhaps the most evolved and most intelligent among all animals, still biologically speaking they are only animals. Frequently, the difference between humans and some other animals is only that of degree. For example, let us consider human intelligence, which is the most important difference between humans and other animals. All multi-cell animals have some degree of intelligence. Same holds true for other abilities and features like dexterity of hands, or speech. Many other animals have these features but not evolved or developed to the same degree as in humans.
So, we may say that whatever qualities humans have, are also present in other animals also, although to a different degree. But if we define human qualities as the qualities or a combination of qualities only humans possess, then by definition no other animal can have the same qualities.