Humans are not animals according to Descartes’ distinction of res cogitans and res extensa. They are also not animals according to many Christian theologians (Jesus came to save humans, not animals). Perhaps humans are not (only) animals also according to the Aristotelian definition of human beings as “rational animals”, which attributes to humans alone a distinctive character. Humans are also quite different than animals when it comes to their respective rights. But here starts a moot point:

from http://www.popsci.com/should-animals-same-rights-people

1. If, in fact, humans have more rights than animals because they are the dominant group, then this resembles very much racism or any other dominion of one group over the other.
2. If, by contrast, humans have more rights than animals because they are different than animals, then what does this difference consist of? If it amounts to rationality, should psychically empaired human beings have no rights?

Since after the end of the Nazi experiments a (more or less) general consensus has been achieved about the fact that psychically empaired human beings deserve the same rights, one is led back either to No. 1 or to a different basis of the human claim for rights. This could be Peter Singer’s claim that one’s moral stand should be calculated not on the basis of one’s ability to reach a soteriological goal or one’s rational value but on the basis of one’s ability of experiencing pain (Singer 1975). This includes psychically empaired human beings. But it also includes at least many animals (one might argue about the fact that many invertebrates with no nerve ganglia cannot literally speaking experience pain).

The discussion about the inclusion of animals within the realm of beings to whom human rights can be ascribed, thus, seems to hit a nerve in Western thought. It seems that no straight line can be legitimately drawn to separate animals and humans and that there is more a net of family resemblances than a straight opposition between the two groups (a dolphin or a gorilla, just to take an obvious example, seem to me to resemble a human being much more than they resemble an amoeba, although all three can be used for the sake of medical research or kept in zoo-like institutions).

The situation is slighly different in other traditions of thought. In Classical Chinese Confucian philosophy, for instance, the idea that we have stronger obligations towards the members of our extended family and towards further “proximate” people is a viable option and one could easily extend this model to animals, so that it would be legitimate to attribute rights first to the members of our families, then to members of our communities, then to further human beings, then to pet-animals, then to further animals with whom we are somehow connected and only at last to further animals. However, this option clashes with the Western ambition of building a universal ethical system, does not it?*

I wrote about Indian reflections on this topic in a forthcoming article (a preliminary draft of which is available here), where I basically argue that most Indian thinkers seem to see non-human and human animals along a hierarchical sequence with no brisk interruption.
Daya Krishna connects this with the utilitaristic approach to knowledge which characterises most Indian explicit reflections about it:

The usual Indian analysis is centered around the hedonistic view of human nature which sees it as naturally seeking pleasure and avoding pain and has a pragmatic view of knowledge which sees the `truth’ of knowledge in terms of its ability to avoid pain and afford pleasure to the humanking. But on this view no distinction is possible between the human and the animal world as the latter also is supposed to seek pleasure or avoid pain and `sees’ the `truth’ of its knowledge in terms of the `success’ achieved by it in this enterprise. In fact, the whole learning theory in modern psychology and the training of animals is based on this premiss (2004, p. 237)

Let me just add that Daya Krishna is thinking of the first aphorism in the foundational text of the Nyāya school (NS 1.1.1), where knowledge is linked to the achievement of one’s summum bonum. In another philosophical school, the Mīmāṃsā, animals are also considered on the same level as humans when it comes to the fact of desiring happiness (PMS chapter 6).

* I am grateful to L.E. for having discussed this topic with me. For a critical discussion of the concept of “rights”, see Amod Lele’s discussion here (and in the previous posts). On why I am citing Daya Krishna, see this post. Within Chinese philosophy, on Confucius vs. Mozi regarding the universality of rights see this post by Eric Schwitzgebel.