I remember being taught that if you tell a pilot to expedite climb/descent they should climb/descend like their life depends on it, because it probably does. For this reason alone I (thankfully) have very rarely used it in a radar environment.
However, when working in the tower, the word expedite gets used far more often, particularly when backtracking aircraft on the main runway. It is a way of telling the crew to hurry-up otherwise someone is going to be carrying out a missed approach.