What really sets us apart from our cousins? Self-awareness?

Nope, apes can do this, too.

This video illustrates that our ape cousins also exhibit self-awareness, by recognizing that the reflected image in a mirror actually represents themselves.

Animals that can pass the mirror test:
Humans
All of the great apes
Bottlenose dolphins
Orcas
Elephants
European Magpies

An amazing bit of footage from the National Geographic documentary "Human Ape".