A "Eureka moment," named in honor of Archimedes, has a thoroughly positive connotation. A "gotch moment" may be positive (if gotcha is understood as being equivalent to "I understand [what you mean]") or—at least in some cases—negative (if gotcha is understood as a trap to make someone look bad).
The earliest instance of "gotcha moment" that a Google Books search finds is from Ann Haley-Oliphant, Exploring the Place of Exemplary Science Teaching (1994) [combined snippets], and it is intended in an extremely positive sense there:
Every year, I set out to share my scientific understanding with my students. On my walls, there are three large statements that do not change from year to year. They are "Communication," "Stewardship," and "Yes, I Can!" With these as my guidelines, my goal is to have each student, at least once, come up to me in a hurry, excited, and say, "Mrs. Freedman, Mrs. Freedman, Mrs. Freedman, you have got to see this! See what I did! Can you believe what I did!" I call this the "gotcha" moment. I work for these moments all year long. They are my real paycheck. They go in the keeper file. I know that, at that moment, this young person is mine, totally and completely. "Carpe momento."
The negative (or at least schadenfreude) sense of the expression is evident in, for example, Book Digest Review, volume 93 (1997) [snippet]:
REVIEW: N. Y. Times Book Rev. p24 N 10 '96 Emily MacFarquhar (800w)
“There is more derring-do than fact finding as Mr. Wu ventures back into the belly of the beast, but he does succeed in providing material for a prize-winning '60 Minutes' segment with a 'gotcha' moment. This comes when an exporter reveals the secret of quality control in prison factories: timely beatings. ... For all the international controversy surrounding Mr. Wu's exploits in China, "Troublrmaker" is not as gripping as 'Bitter Winds' his 1994 ...
And from Christopher Hitchens, Unacknowledged Legislation: Writers in the Public Sphere (2000):
How well I remember the sweetness of it. I was sitting on a plane, reading a copy of The Spectator. Taki Theodoracopulos had written one of his 'High Life' columns, taking ooff after 'the Sodomites of the Big Bagel' (as he thoughtfully described the gorgeous mosaic of gay life in Manhattan). It was a spirited piece, made no less so by my growing conviction that I had read it before. By the time the plane landed, I had attained the 'Gotcha' moment that is the Nirvana of the vengeful reader. It took me only an hour to get a fax of the 'original article' (by Norman Podhoretz. in the New York Post a few weeks previously), compare the sentences word for word, and fire off an extremely contented and self-righteous letter to the editor of The Spectator. I knew that he would have no choice but to print it, and Taki and I were not getting on very well at the time, so it was one of the most chore-free pieces of unpaid riting that I have ever done.
And again in Newsweek, volume 133 (1999):
Haltingly, Boies, a rumpled lawyer, refers to "Exhibit Number 1847A," an e-mail message from the Microsoft staffer that is copied to Chae. "Brad mentioned ... new concerns" about the hearings, the message says. Boies looks up: "Is that Brad that's referred to there you, sir?" he asks politely. In a corner, Chase answers: "I can't be positive, but I assume it was me." Score one for the Feds.
It was another quiet gotcha moment for Boies, whose consciously unflashy courtroom style is turning the Microsoft trial into a clash between the concede-nothing culture of Redmond and a lawyer with the understated canniness of a courtroom Columbo.
A "gotcha moment" clearly has something important in common with a "Eureka moment": the flash of insight and understanding. But in many cases it is achieved in service to an undeclared motive, which is to capture or defeat an adversary. From the perspective of society, this may be viewed as a positive thing or a negative thing, but it is more likely to be perceived as negative when the tactics by which it is achieved border on or cross over into entrapment.