az usually have a more negative meaning than biraz.
Az şeker var. - There is little sugar.
Biraz şeker var. - There is some sugar.
however with adjectives we USUALLY use biraz.
Doktor az yakışıklı. - this is really NOT a common/right way of saying. but it means like the doctor is almost ugly.
Adding on to the deactivated user's comments, this phrase, as it has been translated into English is a bit awkward. Native English speakers are far more likely to write and say "somewhat handsome" rather than "a little bit handsome" and if Tatoeba is any indicator, "somewhat" is an acceptable translation for "biraz." I also typed it in as an answer and duolingo accepted it as a correct translation.