What is the difference between a hydrostatic and a hydraulic system?

By "Hydrostatic" systems one tends to mean "closed loop hydraulic systems". That is hydraulic systems comprising a pump and motor hwere the pump supplies the motor whose return side supplies the pump. The speed and direction of rotation of the hydraulic motor being changed according to the displacement angle of the variable displacement axial pison pump.

"Open loop hydraulic systems" on the other hand comprises various motors often one or more "linear motors" (hydraulic cylinders). When using hydraulic cylinders in hydraulic systems one needs a reservoir to be able to hold the various volume need of the cylinder. (the displacement on the piston side is usually bigger than the rodside).

The reservoir needs a "breather" due to the varous displacement needs hence the "open system".