The branch of anthropology that is concerned with discovering what makes human beings different from other living organisms and what human beings share with other members of the animal kingdom is called

The correct answer is biological anthropology, or physical anthropology.

It is centered around human beings only, with respect to their common or different features that they either share or don't share with their related non-human primates and other ancestors. It deals with human evolution, the study of their behavior, reproduction, etc.