How does PWM control a hobby servo, and how is that different from LED dimming?

A standard hobby servo is commanded by pulse width, not by average voltage. It expects a pulse roughly every 20 ms (50 Hz) whose high-time encodes the target angle, typically about 1 ms for one extreme, 1.5 ms for center, and 2 ms for the other extreme, and the servo's internal controller drives its motor until its position matches the commanded pulse width. So the information is in the duration of the high pulse, and the duty cycle is low (1-2 ms out of 20 ms) and not meant to be averaged. LED dimming is the opposite use of PWM: there the average is the signal, duty cycle directly equals perceived brightness, and you run at a high frequency (hundreds of Hz or more) specifically so the eye integrates it into a steady level with no flicker, with the exact frequency unimportant as long as it's above the flicker threshold. So both use a timer's PWM, but the servo cares about absolute pulse width at a fixed 50 Hz, while the LED cares about the averaged duty at an arbitrary high frequency.