Difference between engine rev up and run-up
In general English usage, including in aviation, "rev up" is a verb form describing the act of increasing the speed of something.  "Rev up the engine" just means give it a shot of power to make it go faster.

"Run up", as used in this case, is actually a noun, describing a procedure to test an engine against go-no-go criteria.  "Do an engine run-up" means perform the run-up procedure, which may include accelerating the engine to a specific speed to put things under a bit of stress, and testing various systems while at speed to detect degraded or failed components or subsystems that are revealed only when under stress.