Difference between a controller (e.g PD) and a compensator(e.g Lead)?
In simply words:

Regulator (controller) is a device that controls the object in closed loop on the basis of difference (error of regulation) between measurements of object's output and external steering signal, regulator tries to reduce error to zero.

Compensator only change object characteristics (transfer function), e.g. phase of output signal. Compensators are used in connection with regulators, in feedback loop (change characteristics of object's output signal) or in steering path (change steering signal from controller to object).

More here: https://en.wikibooks.org/wiki/Control_Systems/Controllers_and_Compensators