Angst can create compelling drama, character development, and interesting psychology when done well; many acclaimed works of art use it to a degree and the appeal of The Woobie is based on it. But like all other good things, angst can be overdone or done poorly.
Wangst, a portmanteau of "whiny" and "angst," (or "wimpy" or "wah" or "whinging"... or possibly "wanker" or "wang" in especially irritating cases) is essentially angst gone wrong. The intended Woobie's Appeal to Pity ends in an epic failure, and thus becomes a pathetically whiny character who insists on crying (often loudly and repeatedly) about a Dark and Troubled Past instead of, you know, trying to move on with their life. This is made even more annoying when the character's "trauma" doesn't come across nearly as tragic as they seem to think it does, compared with other people out there trying to endure yet still determined to solve the problem, making the character's lamentations seem way out of proportion.
Note: Though Wangst is most frequently associated with characters whining over petty "tragedies," the trope is not merely sadness or angst: it is poorly written sadness or angst that comes off as unjustified due to lack of an explanation, exaggeration of the problem, or otherwise not showing an appropriate level of sadness compared to the tragedy in question. It's not necessarily the scale of the tragedy that the character is reacting to that's the problem, but the way it's handled. Note that it can be intentional, and frequently is; when used correctly, it can create or aid and abet schadenfreude, establish a character's childlike or emotionally immature or unstable qualities, or otherwise reinforce the characterization of an inveterate whiner whose failure to place their problems into perspective or see their own role in them is a central flaw.
It all boils down to quality and personal tolerance level; what is grating and unrealistic to one viewer can be genuinely heart-rending for another. There's an inherent Double Standard involving this trope — people are much more likely to be crying "Wangst" to a male character than they are a female character. In heavily action-oriented works, it's expected for anyone who just pauses for a short time to feel a little angst to immediately get called out as wangsty: see also What Measure Is a Non-Badass?.
Wangst is a favorite of Emo Teens, Sympathetic Sues and characters Cursed with Awesome. Any kind of Despair Speech (especially by nihilists) is also often considered as this. Often pointed out by playing the World's Smallest Violin. See also Emo, Narm, Angst Dissonance, First World Problems, Misery Poker, Prone to Tears. To deal with the wangsty kind, people will usually say "Get a Hold of Yourself, Man!!"
The "wangst" equivalent for entire stories is when Too Bleak, Stopped Caring is caused by too much True Art Is Angsty. Silly Rabbit, Cynicism Is for Losers! is the philosophical equivalent. Compare Deus Angst Machina, which is about having too many sources of angst for Willing Suspension of Disbelief rather than too much response to sources of angst. For both in-universe and meta-examples, see Muse Abuse, a frequent Real Life and in-universe cause of Wangst. Contrast Angst? What Angst?, and Stoic Woobie. Compare First World Problems and Misery Poker.
This just defines the term. No examples, please.