The difference between human and other animal communication

In his book The Language Instinct, (1994) Steven Pinker pointed out two fundamental facts about human language that were used by linguist Noam Chomsky to develop his theory about how we learn language. The first is that each one of us is capable of producing brand new sentences never before uttered in the history of the universe.

This means that:

[A] language cannot be a repertoire of responses; the brain must contain a recipe or program that can build an unlimited set of sentences out of a finite list of words. That program may be called a mental grammar (not to be confused with pedagogical or stylistic "grammars," which are just guides to the etiquette of written prose.)

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Did the brain increase in size to cope with rising language ability or did the increasing use of language drive brain development?
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The second fundamental fact is that children develop these complex grammars rapidly and without formal instruction and grow up to give consistent interpretations to novel sentence constructions that they have never before encountered. Therefore, [Chomsky] argued, children must be innately equipped with a plan common to the grammars of all languages, a Universal Grammar, that tells them how to distill the syntactic patters out of speech of their parents. (Pinker, p. 9)

Children have the ability to produce much greater language output than they receive as input but it is not done idiosyncratically. The language they produce follows the same generalized grammatical rules as others. This leads Chomsky to conclude that (quoted in Pinker, p. 10):

The language each person acquires is a rich and complex construction hopelessly underdetermined by the fragmentary evidence available [to the child]. Nevertheless individuals in a speech community have developed essentially the same language. This fact can be explained only on the assumption that these individuals employ highly restrictive principles that guide the construction of grammar.

The more we understand how human language works, the more we begin to realize how different human speech is from the communication systems of other animals.

Language is obviously as different from other animals' communication systems as the elephant's truck is different from other animals' nostrils. Nonhuman communication systems are based on one of three designs: a finite repertory of calls (one for warnings of predators, one for claims of territory, and so on), a continuous analog signal that registers the magnitude of some state (the livelier the dance of the bee, the richer the food source that it is telling its hivemates about), or a series of random variations on a theme (a birdsong repeated with a new twist each time: Charlie Parker with feathers). As we have seen, human language has a very different design. The discrete combinatorial system called "grammar" makes human language infinite (there is no limit to the number of complex words or sentence in a language), digital (this infinity is achieved by rearranging discrete elements in particular orders and combinations, not by varying some signal along a continuum like the mercury in a thermometer), and compositional (each of the finite combinations has a different meaning predictable from the meanings of its parts and the rules and principles arranging them). (Pinker, p. 342)

This difference between human and nonhuman communication is also reflected in the role that different parts of the brain plays in language as opposed to other forms of vocalization.

Even the seat of human language in the brain is special. The vocal calls of primates are controlled not by their cerebral cortex but by phylogenetically older neural structures in the brain stem and limbic systems, structures that are heavily involved in emotion. Human vocalizations other than language, like sobbing, laughing, moaning, and shouting in pain, are also controlled subcortically. Subcortical structures even control the swearing that follows the arrival of a hammer on a thumb, that emerges as an involuntary tic in Tourette's syndrome, and that can survive as Broca's aphasic's only speech. Genuine language . . . is seated in the cerebral cortex, primarily in the left perisylvian region. (Pinker, p. 342)

Rather than view the different forms of communication found in animals as a hierarchy, it is better to view them as adaptations that arose from the necessity to occupy certain evolutionary niches. Chimpanzees did not develop the language ability because they did not need to. Their lifestyles did not require the ability. Humans, on the other hand, even in the hunter-gatherer stage, would have benefited enormously from being able to share kind of detailed information about plants and animals and the like, and thus there could have been an evolutionary pressure that drove the development of language.
Human language was related to the evolution of the physical apparatus that enabled complex sound production along with the associated brain adaptations, though the causal links between them is not fully understood. Did the brain increase in size to cope with rising language ability or did the increasing use of language drive brain development? We really don't know yet.
The argument against a linguistic hierarchy in animals can be seen in the fact that different aspects of language can be found to be best developed in different animals.
The most receptive trainee for an artificial language with a syntax and semantics has been a parrot; the species with the best claim to recursive structure in its signaling has been the starling; the best vocal imitators are birds and dolphins; and when it comes to reading human intentions, chimps are bested by man's best friend, Canis familiaris. (Pinker, PS20)
It seems clear that we are unlikely to ever fully communicate with other species the way we do with each other. But the inability of other animals to speak the way we do is no more a sign of their evolutionary backwardness than our nose's lack of versatility compared to the elephant's trunk, or our inability to use our hands to fly the way bats can, are signs that we are evolutionarily inferior compared to them.
We just occupy different end points on the evolutionary bush.