Difference between compensation and using a PID contoller?
It can be said that PID is a special case of combination of lead lag controller. So a combination of lead-lag will do what a PID can but for a different combination of lead-lag, single PID will not be able to satisfy the condition.

Although sometimes two or three PID (with different values) are used in series rather than designing a new controller as it is simpler. Here is the link for limitation of PID ... though I guess using multiple PID's can achieve whatever we want.