The dance scene: Random, cute girl comes up to guy and pulls him onto the dance floor despite (feeble) attempts to resist; guy stumbles around making a complete idiot of himself for the better part of the song; and finally, guy gets over his nervousness to finish the dance perfectly, just in time for the lights to dim romantically and fireworks to go off!
It also plays as very bittersweet, because you can see Squall sort of smiling near the end, and then she just runs off, which changes to a sort-of disappointed half-frown. Squall's facial expressions here are pitch-perfect. The game only gives us those few movements to work with, but we know how utterly crushed he is on the inside. It's also a sort of mini-downer ending, because considering future reveals in the game, those moments play perfectly into Squall's nihilistic philosophy. All of this, of course, sets up more heartwarming later on though.
An entry on Fridge Brilliance makes this even more Heartwarming. The idea being that Rinoa saw a glum-looking student standing alone with a fresh scar on his face. And she went up to him to tell him he was the best looking guy there and tried to make sure he had a good time.
The final balcony scene, and the kiss you waited the whole damn game to finally see. The best possible note to end the game on.
The inclusion of the (slightly tweaked) original Final Fantasy theme was the icing on the cake.
Squall's adorable smile. The only time he smiles in the game, and it's a damn knockout.
Aside from all that, there is a bit of a scene with Laguna and Raine. At about 2:30 in the above video, you can see Laguna revert to his younger self and getting to live one last moment with her as time unravels into its original form: the moment they were married. It is unclear given the nature of time compression if he was truly there or merely remembering the experience at Raine's grave, but given his initial look of surprise, it would appear to be the former.
The "Garden Square" message board, shut down after one Flame War too many, is re-opened after Squall is made Commander. It's full of good luck messages for him.
Extra meaning if you're a gamer and a 'Net dweeb. And, let's face it, anyone looking at this page not matching that description?
Ellone talking to Squall on the Lunar Base about his trips to the past at her instigation: "Thanks to you guys, I was able to see how much I was loved. I couldn't change the past but just seeing it was more than enough."
The scene where Squall leaves the garden and heads down the railroad on foot carrying the unconscious Rinoa. Also, when his friends catch up with him soon after.
To clarify: Balamb Garden was docked at Fisherman's Horizon, a settlement smack-dab in the middle of the ocean. Railroad tracks extend from either side to land at separate continents. That's right: Squall carried Rinoa on his back halfway across the ocean on the slim chance of finding a hidden civilization, on the even slimmer chance that they might be able to help her.
Dr. Kadowaki: Talk to your men, Squall. Encourage them. As their leader, it's your duty.
Nida: You probably don't know, but everybody in this Garden looks up to you. They like you.
Squall: ......Everybody, this is Squall. How's everyone doing? You're all probably too tired to even stand up after all the fighting. But I want everyone to listen to me... We still have a chance to win, and I need your help. This is going to be our final battle. We're going to attack them before they come in again. To do that, we're going to head straight into their Garden. So I want everyone to prepare for a major collision. Take care of the junior classmen. Irvine, Quistis, Zell, and Selphie will lead the attack into their Garden. As for everyone else, please support them if you can. SeeD was formed to fight the sorceress: at least, that's what I heard. And Garden was created to train SeeDs. So this battle is Garden's destiny and also our destiny. It's a grueling battle, and I'm sure you guys are all exhausted. But I don't want to have any regrets. I don't want anyone to look back and regret this day. So just this once, I want you guys to give everything you got! For yourselves and for me!
His heroic rescue of Rinoa from certain death shortly afterwards actually makes the infuriating minigame worthwhile.
When Squall frees Rinoa from being sealed away by the Esthar scientists. She rushes to his arms, and they hug. D'aaaaw. :D
When Squall tells Rinoa, "Even if you become the world's enemy, I will be your knight".
The promise he makes to her afterwards. "I'll be here...waiting for you. You'll find me."
Seifer's final scene managed to sell his redemption perfectly. The genuine joy on his face from his friendship with Raijin and Fujin, as well as his expression upon seeing the Garden, are powerful.
Disk 2's Great Escape sequence includes a moment in which Squall leaps down several stories to save Zell's life. Once they're free, cue Selphie and Quistis teasing Squall going through those lengths to save Zell.
During the preparations for the concert in Fisherman's Horizon in Disc 2, the player very briefly controls Irvine to select the music for the event. It's possible to leave the stage area and wander around FH and Garden with him, during which he can have a confrontation with the same FH resident who gave Squall a hard time on their arrival:
Familiar Face #3: That's strange... You don't look like one of them... So why are you with them?
Familiar Face #3: Those warmonger SeeDs.
Irvine: Before, I was just a solo sniper... A lone ranger. I was alone in my battles: my battles against pressure, my battles against death. It almost drove me nuts at times. And it was 'them' who saved me from it all... 'They' are what you call comrades.
Familiar Face #3: Then you are one of them. Another warmonger.
Irvine: Yup... I guess so. I'm a warmonger. And you... judging us... Who are you?
It was mentioned in passing, but the scene onboard the Ragnarok: Rinoa fails (supposedly at the risk of her own safety) to resist the temptation to jump into her beloved's arms. It's adorable and sweet. So much so that Squall responds visibly to being caressed so lovingly, and, whilst still being unable to put his feelings into words, finally returns the favor with his own adorable hug.
The scene where Squall tells Mayor Dobe of FH his thoughts on war and peace:
Squall: I wish you could be a little more understanding about us. We're not just a bunch of warmongers.
Mayor Dobe: Oh?
Squall: It's hard for me to explain... I wish...everything could be settled without resorting to violence... ...and there would be no need for battles. Like you've been preaching, it would be wonderful if things could be settled by discussion. The only problem with that is that it takes too much time. Especially if the others are not willing to listen. So I believe that fighting is inevitable at times. It's really sad. That's all I have to say. I hope you understand someday. I think the world needs both people like you and people like us. Thank you for all your help. Good bye.
At the end of the game, Edea seeing the mortally-wounded Ultimecia and knowing that she cannot die without passing on her power, willingly takes it upon herself, knowing full well that ending Ultimecia's agony and protecting one of the orphans from being forced to take the power themselves will mean the end of her own happy life.
After you beat the Fisherman's Horizon boss, you see the party that was sent to destroy the missile base climb out of the wreckage dazed and a little bruised and battered, but none the worse for wear, even after their brush with death at the base. Squall is the first to greet them. His reaction? A display of genuine excitement and happiness. In fact, if it wasn't the fact that he likes to keep his distance, he might have given them a hug. Double points if Rinoa was one of the people you sent.