I can't answer your question completely, but I can confirm that "Triefnase" is probably much rarer and more ambiguous than "Rotznase", which should be relatively universally understood.
A guess toward the different meanings:
"Triefnase" is a slouch. It's not hard to see the connection: People with runny noses usually have a cold, which makes people slow (both in the literal sense as well as in the figurative one).
"Rotz" or "Rotze" is a relatively vulgar word for snot, and is used often in a more active negative sense ("Auf die Straße rotzen" invokes a stronger image than "auf die Straße spucken").
"Rotznase" (synonym: "Rotzlöffel"!) carries a rougher, more repulsive and more derogatory connotation than just "somebody with a runny nose"."Rotz" can be translated as "shit" in a sentence like:
Das ist totaler Rotz!