Collecting a golden stopwatch triggers a Time Trial in most games it's featured in. The stopwatch only appears in levels where the crystal has already been collected. Completing a Time Trial quickly translates into a relic as a reward. Some elements are changed during Time Trials: all Wumpa fruits are removed, there are no checkpoints, and some crates turn into time crates that freeze the timer for a few seconds when broken.
In Mind Over Mutant, stopwatches trigger optional, timed mini-games, such as collecting green Mojo or breaking all the crates.