From The 2% Difference, an article by Robert Sapolsky:

Given the outward differences, it seems reasonable to expect to find fundamental differences in the portions of the genome that determine chimp and human brains—reasonable, at least, to a brainocentric neurobiologist like me. But as it turns out, the chimp brain and the human brain differ hardly at all in their genetic underpinnings. Indeed, a close look at the chimp genome reveals an important lesson in how genes and evolution work, and it suggests that chimps and humans are a lot more similar than even a neurobiologist might think.

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... Still, chimps and humans have very different brains. So which are the brain-specific genes that have evolved in very different directions in the two species? It turns out that there are hardly any that fit that bill. This, too, makes a great deal of sense. Examine a neuron from a human brain under a microscope, then do the same with a neuron from the brain of a chimp, a rat, a frog, or a sea slug. The neurons all look the same: fibrous dendrites at one end, an axonal cable at the other. They all run on the same basic mechanism: channels and pumps that move sodium, potassium, and calcium around, triggering a wave of excitation called an action potential. They all have a similar complement of neurotransmitters: serotonin, dopamine, glutamate, and so on. They're all the same basic building blocks.

The main difference is in the sheer number of neurons. The human brain has 100 million times the number of neurons a sea slug's brain has. Where do those differences in quantity come from? At some point in their development, all embryos—whether human, chimp, rat, frog, or slug—must have a single first cell committed toward generating neurons. That cell divides and gives rise to 2 cells; those divide into 4, then 8, then 16. After a dozen rounds of cell division, you've got roughly enough neurons to run a slug. Go another 25 rounds or so and you've got a human brain. Stop a couple of rounds short of that and, at about one-third the size of a human brain, you've got one for a chimp. Vastly different outcomes, but relatively few genes regulate the number of rounds of cell division in the nervous system before calling a halt. And it's precisely some of those genes, the ones involved in neural development, that appear on the list of differences between the chimp and human genomes.

That's it; that's the 2 percent solution. What's shocking is the simplicity of it. Humans, to be human, don't need to have evolved unique genes that code for entirely novel types of neurons or neurotransmitters, or a more complex hippocampus (with resulting improvements in memory), or a more complex frontal cortex (from which we gain the ability to postpone gratification). Instead, our braininess as a species arises from having humongous numbers of just a few types of off-the-rack neurons and from the exponentially greater number of interactions between them. The difference is sheer quantity: Qualitative distinctions emerge from large numbers. Genes may have something to do with that quantity, and thus with the complexity of the quality that emerges. Yet no gene or genome can ever tell us what sorts of qualities those will be. Remember that when you and the chimp are eyeball to eyeball, trying to make sense of why the other seems vaguely familiar.