On the serious side:both SALOON and SALON can have other precise meanings, but inthe context of this discussion:
a SALOON is the (stereo)typical name for the local pub in the Wild West
a HAIR SALON (maybe at a time you could say saloon, but not normally now) is a place where all kinds of hair services (cuts, washing, perming, coloring etc) are done.
a BARBER SHOP is generally geared toward male clients and specializes in shorter haircuts and shaving. The word "barber" has to do with the latin word for beard - "barba".
Some places may be called "mens' hair styling" or just "hair styling" but seem to be more barber shops than salons.