I'm not clear on them. How are they treated differently, if at all?

Additionally, what qualifies as an emulator? I know Dolphin is one, but what about playing on the Wii?

What about the Wii U (through Dios Mios/Nintendont)?

I've noticed that some runs on here with the emulator box checked have RTAs on this site that are different from the timer in their videos. Why is this?

I would appreciate any information on this.