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One discards the depressurized hydraulic fluid, the other recirculates it back to the pump. That's all there is to it.
For example, the Falcon 9 rocket first stage uses open loop, using pressurized nitrogen to pressurize the hydraulic fluid to control the grid fins, then directs the depressurized fluid that leaves the actuators into the fuel tank and uses it as fuel during landing.
In closed loop hydraulic system motor return is connected directly to the hydraulic pump inlet and in open loop the hydraulic pump inlet and the motor are connected to the reservoir. If you are interested to find more details on this, check: Types of hydraulic system