Balance drum is a rotating component with a close clearance wear ring (to change thrust + or -.) Changing the diameter (area) of the drum determines how much it will affect thrust, differential pressure across the drum x area of the drum. It can be used in vertical pumps.
Balance line is piping, to equalize pressure in different areas of the case. It can be used in multistage between bearing horizontal pumps, so thrust is neutral or reduced as much as possible.
The balance line usually refers to the pipe that routes the leakage from the balance drum or disk back to suction. There is an FAQ in this forum on the subject that may help.
