Can anyone tell me the difference between a nudnik and a nudzh?

Well, there both from the same Slavic root (probably Polish nuda 'boredom'): nudnik is a nomen agentis, 'one who bores, or bothers' from nudyen 'to bore, pester' (the digraph {dy} in Yiddish indicates a palatalized consonant). The suffix -nik also comes from Slavic, rather than the Germanic base of Yiddish. The verb nudzh (also nudge) is used as a noun also, probably under English influence: words changing syntactic categories without derivitive morphology, i.e., no suffixes needed. You see the same thing with the English noun schmier (for cream cheeese, etc.) derived from the Yiddish verb shmiren 'to smear'. The PIE root *n@u-ti- also gives English need and German Not.

So, what's the diff between 'em. Not much, I'd say.

I understand a nudnik to be a pest and a nudge, a pesterer. One is an annoyance and the other, an annoyance who wants something out of you.

Dear IP: Could it be that a nudzh is actively annoying,
and a nudnik is only passively annoying?

<<..passively annoying>>

wwh: Dismissing the 'irony' of psychologisms (passive aggressive) as precedent, wouldn't you say that to act passively was paradoxical?

Dear IP: There are those who annoy by being stupid, but
may say or do very little. Then there are half-wits who
butt in with stupid remarks.

Perhaps, the innocent bore versus the malicious? I took a look at Harkavy's. He gives a noun nudy (i.e., nudzh) that is glossed as nudate 'nausea'. The female nudnik is a nudnitse, and the daytshmerish (German, 'fancy word') for 'tedium' is langvaylikayt (langvaylen 'to weary').

die Langweile = boredom, verb langweilen = to bore

Yep, Yiddish is a Germanic language like English, Dutch, and German.

die Langweile = boredom

Thanks for reminding me of one of my favorite words from German class -- when you're bored, things to seem to take a Long While!

A couple my parents knew years ago dubbed their unborn child Nudnik. After he was born, the nickname stuck -- I think until he was a teenager.

A definition from "yiddish For Yankees":
nudge (nooj) To bother. To make a nuisance of yourself. Your husband is enjoying himself with a good book. You NUDGE him about vacation plans for 2012, he shouldn't be too content.

No-one's mentioned a phudnik yet: a nudnik with a Ph.D.

Dear Jenet: I wish I had known "phudnik" forty years ago
when I had to work with some of them in VA psychiatric
wards.