A positive displacement pump is a pump that moves the same amount of liquid per rotation regardless of the speed of the rotation . . . . . . There, clear as skimate?
The 2 main types of pumps are positive displacement and centrifugal. The majority of pumps used in the fish keeping hobby are centrifugal. A centrifugal pump does not have a sealed pumping chamber, it is open. The impeller spins and the blades in the impeller throw the water away from blade by centrifugal force. A centrifugal pump is a variable torque load, meaning that the faster you turn it, the more water it moves with each spin, the more torque is required to spin it.
A positive displacement pump is a constant torque load. Positive displacement pumps have a sealed pumping chamber, such as a piston or a progressive cavity. One full turn of the pump will move the same amount of water each time it spins, regardless of how fast it spins so you need full torque on the motor regardless of the speed. A centrifugal pump will move more water with each spin, as the RPM increases, to a point. Also, a centrifugal pump will only create a certain amount of head pressure, even if the output is deadhead. A positive displacement pump will continue to pressurize against a deadhead until either the pump or the plumbing fails, explosively. You have to provide overcurrent protection for a positive displacement pump to shut it down if a deadhead situation occurs.
Also, if a centrifugal pump is spun too slowly, around 20-25 Hz or below, it will not create enough centrifugal force to move any water.
My guess as to why they are not used more often would be complexity. Positive displacement pumps usually have multiple moving parts are therefore are more likely to have mechanical problems.