For example, I looked at Spyro The Dragon runs and saw that some EPSXE versions are allowed and others are not. Is this because some emulators give unfair advantages and others don't? Does it have something to do with the speed?
There are lots of things that an emulator can do inaccurately like load times, handling lag, down to being inaccurate at calculations and basic game speed. From what I have crudely compared so far as an example VBA handles lag in Pokémon Emerald worse than an actual console does and plays the game at about 99.5% speed while I'm streaming (still shows as 100% and the correct framerate, so it's 99.5% or better). Usually the most accurate emulator will get picked as the preferred emulator, unless there are multiple that play the game perfectly of course. EDIT: I want to also note that being slower is as bad as being faster, it's just unfair towards different players.