Classical Chinese distinguished facial hair by location.
髭 is specifically the upper-lip moustache. The full taxonomy:
髭 (moustache),
须 (chin beard),
髯 (cheek whiskers / sideburns),
髦 (long hair forelock). In modern writing the precise sense has blurred — the common compound
髭须 just means 'beard / moustache' generally. The character is high-register; everyday Mandarin uses
胡子.