The difference is that how much weight they lift is a side effect - the real goal is appearance. Unlike, say, Olympic power lifting.
Olympic weight-lifting is constituted of the snatch and the clean and jerk. Power-lifting is normally constituted of the "big three" lifts: bench press, squat, and deadlift.
There is also an informally defined idea of "power-building"[3] which is sort of a cross between bodybuilding and power-lifting.
And then there's crossfit, and strongman, and ... Anyway, enough said about all of that. :-)
[1]: https://en.wikipedia.org/wiki/Powerlifting
[2]: https://en.wikipedia.org/wiki/Olympic_weightlifting
[3]: https://www.muscleandfitness.com/routine/workouts/workout-ro...