Well, in the past "beard" usually meant the "hair that grows on a man's face often" but not necessarily "excluding the mustache" (Merriam Webster), and whisker meant a hair of the beard, ergo whisker=facial hair. The other terms seem like they were used when you shaved part of the beard away and left some subsection, e.g. sideburns, muttonchops, mustache, circle beard, goatee, what have you, but it was all part of the beard.
Well, sure, but when you're talking about the wide variety of styles, you wouldn't say "beard." A mustache isn't a type of beard. Sideburns aren't a type of beard.
Whiskers just sounds...dumb. Cats have whiskers. If you were to describe a man's facial hair as "whiskers," I'd picture something very thing and scraggly coming out of his cheeks.