Before I started painting portraits, I wondered what was the big deal. But after painting a few, you begin to understand "the big deal."
First, faces are the first visual stimulus that capture an infant's attention. That connection with faces must persist, even as we grow older and are distracted by other shiny things.
Also, the artists that do the best portraits can capture the depth in a face, will also balancing the symmetry and asymmetry inherent to each face. And we aren't even talking about the basic expression that belongs to each person. You cannot capture the expression until after you have the other less tangible piece in place.
Now I view portraits as problem solving made fun. My Myers-Briggs type says I am a "natural born problem solver."