From another thread on the board:
"Hunters are judged subjectively on suitabilty to be a field hunter, on jumping form, style, manners and way of going.
Jumpers are judged objectively on number of rails down and disobediences, usually converted to faults, and speed to complete the course. Form, style, manners and way of going doesn't count."
Hunter under saddle, to me, means a flat class, as a distinction from hunter over fences. But to my chagrin I found, in the thread I quoted myself from above, there are regional differences in what classes are called.
I would guess hunter under saddle started as a distinction from Conformation Hunter, which is a halter class.
There's some subtler differences - hunters are supposed to be shown over fences simulating natural obstacles - gates, coops, etc.; jumper classes have all the striped poles and brightly colored stuff. Hunter riders are usually very conservatively dressed, and there are restrictions on what types of tack can be used. Jumpers have fewer such restrictions.