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I found Noam's hypothesis very compelling in the past. I still think that the idea that language is somehow a cultural or social invention of our species is wrong. But I think that there is a chance (we don't know that, but it seems to most promising hypothesis IMHO) that the difference between humans and apes is not a very intricate special circuit, but genetically simple developmental switches. The bootstrapping of cognition works layer by layer during the first 20 years of our life. Each layer takes between a few months and a few years to train in humans. While a layer is learned, there is not much going on in the higher layers yet, and after the low level learning is finished, it does not change very much. This leads to the characteristic bursts in child development, that have famously been described by Piaget.

The first few layers are simple perceptual stuff, the last ones learn social structure and self-in-society. The switching works with something like a genetic clock, very slowly in humans, but much more quickly in other apes, and very fast in small mammals. As a result, human children take nine months before their brains are mature enough to crawl, and more than a year before they can walk. Many African populations are quite a bit faster. In the US, black children outperform white children in motor development, even in very poor and socially disadvantaged households, but they lag behind (and never catch up) in cognitive development even after controlling for family income.

Gorillas can crawl after 2 months, and build their own nests after 2.5 years. They leave their mothers at 3-4 years. Human children are pretty much useless during the first 10-12 years, but during each phase, their brains have the opportunity to encounter many times as much training data as a gorilla brain. Humans are literally smarter on every level, and because the abilities of the higher levels depend on those of the lower levels, they can perform abstractions that mature gorillas will never learn, no matter how much we try to train them.

The second set of mechanisms is in the motivational system. Motivation tells the brain what to pay attention to, by giving reward and punishment. If a brain does not get much reward for solving puzzles, the individual will find mathematics very boring and won't learn much of it. If a brain gets lots of rewards for discovering other people's intentions, it will learn a lot of social cognition.

Language might be the result of three things that are different in humans:
- extended training periods per layer (after the respective layer is done, it is difficult to learn a new set of phonemes or the first language)
- more layers
- different internal rewards. Perhaps the reward for learning grammatical structure is the same that makes us like music. Our brains may enjoy learning compositional regular structure, and they enjoy making themselves understood, and everything else is something the universal cortical learning figures out on its own.

This is a hypothesis that is shared by a growing number of people these days. In humans, it is reflected for instance by the fact that races with faster motor development have lower IQ. (In individuals of the same group, slower development often indicates defects, of course.)

Another support comes from machine learning: we find that the same learning functions can learn visual and auditory pattern recognition, and even end-to-end-learning. Google has built automatic image recognition into their current photo app:
http://blogs.wsj.com/digits/2015/07/01/google-mistakenly-tags-black-people-as-gorillas-showing-limits-of-algorithms/

The state of the art in research can do better than that: it can begin to "imagine" things. I.e. when the experimenter asks the system to "dream" what a certain object looks like, the system can produce a somewhat compelling image, which indicates that it is indeed learning visual structure. This stuff is something nobody could do a few months ago:
http://www.creativeai.net/posts/Mv4WG6rdzAerZF7ch/synthesizing-preferred-inputs-via-deep-generator-

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