Sometimes I wonder at the uniqueness of each individual's face. It just amazes me how everybody, no matter how numerous we have, has a different face. With the exception of twins and non-familial coincidental twins.
But you probably have one eye slightly higher than the other, one eyebrow that is shaped different, or your nose might turn up ever-so-slightly. Your cheeks might show more "chub" when you smile compared to the next person, even though the weight is similar. I think this is why painted or drawn portraits sometimes don't quite capture someone's face, even though it seems horribly close and is obviously a picture of the subject - and weirdly, why simple caricatures seem to do such a good job. The folks that do them have trained themselves to notice and highlight these sorts of differences.
As a sidenote, I still find it amazing that the human brain is so quick to pick up such things, especially considering it takes training and practice to reproduce on paper.
People have proposed leveraging this to generate unique avatars, since you can just pick some random parameters for the arrangement of a stick figure face and it will probably look unique.