I think that maybe the word I'm looking for is expressiveness. I like expressive faces - faces that can powerfully convey a range of different emotions ... and the different aspects of the person's character.
Tom Cruise is an example (to me) of a handsome but uninteresting face. I mean, there seems to be a rather limited number of expressions he can have on his face.
Interesting faces:

Beautiful too.

Completely unattractive, and yet very interesting.

Too weird to be "beautiful", but oozes quirkiness, originality, emotion.

Boy, is he ugly ... but this is maybe a good example of what I mean about a face that shows a lot about what the person's personality, mood, character etc.