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After hearing how badly the fight is going for the Starfleet forces, Picard announces his intention to violate orders and take the ship into combat to help, but will allow any dissenters to speak up. As funny and awesome as it is, Data's response after the beat is also a heartwarming display of loyalty:
Data: I believe I speak for everyone here, sir, when I say, "To hell with our orders."
Riker helps Cochrane get over his fear of destiny:
Riker: Don't try to be a great man. Just be a man, and let history make its own judgments.
Cochrane: That's rhetorical nonsense. ... Who said that?
Riker: (grinning) You did, ten years from now.
Picard apologizing for calling Worf a coward:
Picard: I apologize for some of the things I said to you earlier.
Worf: (bitterly) "Some"?
Picard: As a matter of fact, I think you're the bravest man I've ever known.
Worf: (Standing up a little straighter) Thank you, sir.
Actually, when you consider that Worf is a Klingon and Picard just called him a coward, his response kind of counts as one too...
Worf: If you were ANY OTHER MAN, I would kill you where you stand...
Picard telling Riker to "stay out of history's way" when evacuating the Enterprise crew to Gravett Island. There's something charming and heartwarming about the scene - probably because they think they won't be able to return home and it's sort of a retirement package to live out their days on a deserted Polynesian island... Wait a minute...
Picard going back for Data as the ship is evacuating because he's certain the android is still alive. He justifies it by recalling how the crew risked everything to save him from the Borg years earlier.
"I owe him the same."
Cochrane's subdued "Oh, wow" at seeing the Earth from space. "It's so small." One wonders if that didn't help, just a little, to turn him into the man he was to become.
The epilogue, where aliens land to find the human who flew Earth's first warp-drive vessel. When the visitors reveal themselves to be Vulcans, you know it is the Dawn of an Era where humanity's time of desolation and despair has ended and its hopeful beginning of the building of The Federation has begun.
Every little moment of wonder from the 21 century Terrans led to a slight dampness of eye, because it's so perfectly what Trek is about—this is what the future could be! And then when the first Vulcan pulls back his hood to reveal his ears, and the music crests, and you know everything that will unfold and everything that is to come...yeah. Definitely something in the eye.
It's also great on in meta sense, many of Gene Roddenberry's older fans have often gone on the record of how Star Trek's creator would not have approved of many of the shows and films that have come out since his death. But just about everyone can agree that he would have liked the idea of the first meeting between humanity and Vulcans.
Enterprise later used this to illustrate how the Mirror Universe diverges, with Cochrane suddenly shooting the Vulcan instead, and it works very effectively as a deliberate Shocking Swerve because of how heartwarming the original scene was.
A more subtle part is the Enterprise crew being unable to resist the urge to observe the moment for themselves. They ought to have just beamed back to the ship... (And Sisko shouldn't have introduced himself to Kirk, etc.)
Picard and Lily's final scene together, as this all unfolds.
Lily: I envy you; the world you're going to. Picard: I envy you; taking these first steps into a new frontier.