The best explanation I’ve seen yet (from an anonymous Wordforum contributor)

The first qui/que (Qui/Qu'est-ce…) and the second one (… est-ce qui/que) play different grammatical roles and indicate different things:
The first refers to the nature of what you are inquiring about: is it a person or a thing? Qui est-ce… is for people and Qu'est-ce… is for things.The second refers to the grammatical function of the unknown person or thing in your question: is it the subject or the complement of a verb? …est-ce qui is for subjects and …est-ce que is for complements.Examples:

Qui est-ce qui fait X ? → Who is doing X?
The first qui indicates that you're asking about a person ("who"), while the second qui implies that the unknown person performs the action of the verb: this person is doing X. 
Short form: Qui fait X ?

Qui est-ce que tu as vu ? → Whom did you see? or commonly Who did you see?
The qui indicates that you're asking about a person ("who" or "whom"), while the que implies that this unknown person is the complement of the verb "to see": the unknown person got seen, and tu is the one who saw them.
Short form: Qui as-tu vu ? (requires inversion)

Note that the English language requires (theoretically, in formal contexts) two different words to ask about people: Who = Qui + qui while Whom = Qui + que.

Qu'est-ce qui fait X ? → What is doing X?
The que (elided to qu') indicates that you're asking about a thing ("what"), while the qui implies that this unknown thing performs the action of the verb: the thing is doing X.
No short form in everyday usage.

Qu'est-ce que tu as vu ? → What did you see?
The first que (elided to qu') indicates that you're asking about a thing ("what"), while the second que implies that the unknown thing is the complement of the verb "to see": tu is the person who saw something, the unknown thing is what got seen.
Short form: Qu'as-tu vu ? (requires inversion)