Does it take special genes to make a special human?

We humans think we're pretty special. Here's Hamlet, in the speech that gave this blog its name:

What a piece of work is a man! How noble in reason! how infinite in faculty! in form, in moving, how express and admirable! in action how like an angel! in apprehension how like a god! the beauty of the world! the paragon of animals!

—Hamlet, Act II, Scene II, The Oxford Shakespeare

Kemble in the role of Hamlet.
Courtesy of Wellcome Images.

It is common, at least in the West, to consider humans "the paragon of animals," typically making reference to the ancient idea of "God's image." I won't address here whether humanity is a paragon, but here's a more tractable question: what are the facets that distinguish humans from other animals? Biologically speaking, what is special about humans that sets them apart from other apes?

Anatomically, we're pretty unremarkable apes. We have enlarged gluteal muscles and other adaptations that make us good distance runners. We have fancy thumbs that make us good tool users. We have a tweaked larynx that facilitates speech. We have spineless penises that facilitate other things. And of course we have what Hamlet was talking about: big brains and associated cognitive abilities.