AUTO is the autothrottle mode, where the throttles are moved automatically to maintain 15 AoA, recommended to use with ACLS.
BOOST is the normal manual operating mode. In this mode, throttle movements are detected by a position senser which then commands an electric motor to set the power lever shaft (which in turn commands engine power) to the desired position.
MAN mode is a backup mode where the power lever shaft is directly controlled via push-pull cables attached to the throttle levers. In the real AC, this mode will cause mechanical hysteresis effects (meaning that moving the throttles from position A to position B and then back to A will not results in the same power setting at position A as before). I don't know if this is modelled though.