Simply put, "Sins of the Father" could easily be known as the true legacy of the third season. On the one hand, it's easy to acknowledge the importance of "The Best of Both Worlds," and on the other, an entire mythology and basically an entire series stems from "Sins."
This is the episode that at last allows Next Generation to expand on the existence of Klingons within its framework. Yes, Worf has been there from the beginning, and yet until this point he was a lot like the Spock of this series, the alien who is alienated from his own culture but nonetheless is a perfect representative of it. Until the J.J. Abrams movie, Spock didn't have a lot of material that went into much depth about his conflicts with other Vulcans (aside from his own father). Worf gets that in spades in "Sins."
Part of his backstory was that Worf was orphaned when he was very young, so that he was raised by humans. His parents were killed during the Klingon/Romulan conflicts that served as the backstory of the entire series (and were a prominent feature of "Yesterday's Enterprise" a few episodes earlier). Of course, early in the series its creators preferred to focus on new aliens rather than established ones. This is why we got the Ferengi, and why it was so complicated to explain their own history with Picard and Starfleet in general. When the Klingons showed up at all, it was mostly to make Worf uncomfortable, or give Riker a chance to showcase his particular skills outside the shadow of his captain. The Romulans showed up at the end of the first season, but pretty much continued the behavior they'd apparently been maintaining in the recent past. Then the third season brought them back to prominence again, showed how complicated it can be when they're involved.
When they're involved with Klingons, it's complicated indeed. In the original series, they were apparent allies. It's only reasonable to assume that eventually one of them decided their interests were better served alone, and thus they went to war against each other. Worf is the most famous victim of this conflict, and "Sins" demonstrates how that is a fact that only gets worse in time.
The episode is a deep immersion into Klingon culture and politics. We meet both Kurn and Duras in the process. Kurn is Worf's brother. The episode begins as Kurn serves the same function aboard the Enterprise that Riker did a season earlier aboard a Klingon ship in "A Matter of Honor." Worf doesn't know he has a brother. This part of the episode is fairly similar to other material in the season. It's when the truth is revealed that things truly become interesting. Suddenly it's no longer just another episode. It's all about Worf, the backstory, everything. Picard even gets a chance to expand on his diplomatic abilities, the thing that truly differentiates him from Kirk. When he gets his hands dirty, it's in the service of getting out and walking around, not shooting or brawling. He seeks out missing pieces of the puzzle, not just telling people that they were wrong to believe this or that.
Anyway, Kurn very quickly becomes a key figure of the entire franchise. He's the version of Worf who remained among his own kind. Yet they're not really so different. Kurn only makes a handful of appearances, but he quickly becomes what Gene Roddenberry would have called a beloved character. As much as Worf himself, Kurn helps inform the depths of the episode, when Duras attempts to hide his own shame by saying it was Worf's father Mogh who betrayed the Klingons at Khitomer to the Romulans, the very place of his death. Better to slander the dead. Kurn has been hiding his true identity. Picard quickly realizes that Worf will not only represent himself but the entire Federation.
All of this pulls us out entirely from the usually episodic material of the series. The series plays at it again a few times, but arguably could very easily have completely changed its format to accommodate this revelation. Deep Space Nine did in fact do this on a regular basis, and eventually embraced Klingons more fully than Next Generation ever did, and even after acquiring Worf didn't stop at him to do this.
Duras later returns and becomes far more infamous. It can be argued that the narrative of Next Generation could be said to belong to a number of different characters. Worf could very easily be one of them. When he murders Duras, or temporarily resigns from Starfleet to help fight the Klingon Civil War, these moments again shatter the format, and if it weren't for the Borg, everyone would think of this grand Klingon saga when they think of Next Generation. Yet even the series lost track of this. By the end it had become an afterthought. Those pesky Duras Sisters? It's like they aren't even related to such a notorious figure.
"Sins of the Father" turns his own people deliberately against Worf, who accepts dishonor, the very thing Klingons abhor, and then returns to Starfleet, where he effectively hides and doesn't really think about any of it except for the periodic occasions where it's too obvious to ignore. Perhaps this is just as well. A moody Worf can be depressing. This is the rare Star Trek character who actually considered suicide, after all (Neelix and O'Brien would be two others). It could still have reshaped the series without going that dark. Instead the Borg delivered the memories everyone has. Sure, most people will call "Sins" a standout, and in a lot of ways it was a permanent game-changer. It went where Star Trek had never gone before. Yet it could so easily have done much more.
Maybe I'm just trying to envision a reality where Kurn was around more often...
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Memory Alpha summary.