Warning: Spoilers Off applies to this page. Proceed at your own risk.
The first cutscene is Noctis in a white void listening to music and acting like he's sitting in a car seat. Then the music fades, he falls over, and looks around. Upon realizing he has no clue where he is, he blames Gladio.
Lightning and the Warrior of Light catch up. Noctis repeatedly tries to get someone to explain this to him and they ignore him.
The description for that cutscene panel afterwards observes that Noctis has no friends, no ride, and no phone.
Shantotto quickly stumbles upon a procession of villains.
Terra puts on a cute pout in an effort to convince Squall to let her, Zidane, and Lightning join him in hunting summons. Squall tells her that her face is gonna get stuck if she keeps pouting like that... and Terra tells him to let them come along so it doesn't. The entire crew steps past him and Squall is left standing there like an idiot.
On route to Odin, Terra picks up Ramuh in another direction. On hearing that Ramuh's closer, he starts heading back towards Odin, clearly just wanting to be alone for a couple seconds... and Lightning goes after him. Zidane and Terra proceed to compare the situation to a mother not wanting her kid to get in trouble.
When Lightning calls off Vaan for not listening to Terra's exposition about Spiritus and Materia, Vaan proceeds to drag Squall into the conversation by proclaiming that he's not listening either. Never mind the fact that Squall and Lightning were already walking with Terra and Zidane when he got there, and have probably already heard the whole thing.
In the Locke/Yuna/Rinoa story update, Locke mentions a serious sort who was insistent that the champions had to complete their mission. His imitated mannerisms imply it was Squall. Rinoa immediately guesses that he was always frowning, saying that she knows the type.
In the middle of a battle, Bartz suddenly realizes that Cloud is named after clouds. As in the "Like puffs in the sky". Cloud says if he remembers that, than he should remember where they are.
In that same fight, Cloud has a quick clash with Sephiroth. This is a huge Red Herring, because NT forsakes the previous games' story format where characters from the same game act as nemeses to each other (with the exception of the Warrior and Garland), and in fact Sephiroth gets more highlights of fighting Lightning. Which means that Sephiroth, who has already heard Cloud say that he's tired of their pointless battles, is officially and deliberatelytrolling him at this point.
Shantotto and Ultimecia banter back and forth in what's practically a rap battle, while Tidus and Firion, and Kefka and Exdeath, just watch in confusion. Also doubles as a moment of awesome for Shantotto, as she manages to make the ever calm and collected Ultimecia lose her cool, while she just keeps on doling out savage rhymes.
In Kam'lanaut's story update, his fight with Onion Knight, Vaan, and Y'shtola takes place on the Floating Continent. Afterwards, the planesgorgers cut off their attempt at fleeing, and a new gate appears... off the edge. Onion and Vaan's reactions are good enough, but then Shtola comes up behind them, tells them to 'have courage', and shoves them.
Kefka, bored with nothing to do, passes the time by sitting atop the Rabanastre palace and casually blasting the city. "Kill me now," he says. Really puts his Orcus on His Throne behavior from his original game into perspective: he was bored and waiting for the heroes to come get him!
How does Shantotto get separated from Firion and Tidus? She's too busy thinking about the gate to the Rift to notice when a portal opens in front of her, and she just falls over likea deactivated robot.
She ends up landing in front of Cloud and Bartz of all people, who were stuck at Alexandria since apparently Bartz kept having them go in circles. Right as he expresses frustration at being lost, boom, Shantotto lands before them and is able to get them out of the area.
Terra and a moogle gang up on Kuja in an effort to get him to come along. Zidane, who has been borderline flirting with Terra... realizes his hopes have been dashed, and decides to guilt-trip Kuja.
Zidane: You're not gonna say no to a lady, are you?
Everyone is constantly exasperated at Vaan not paying attention, wandering off and getting distracted while they're trying to explain things. When Lightning calls him on it, he fires back that Squall isn't paying attention either.
The Warrior of Light, Cecil, and Noctis, are almost to Spiritus' domain when they find out some crucial information to relay to Materia after meeting up with Garland. Rather than walk all the way back, Noctis makes a suggestion: If Spiritus really isn't evil like Garland says, then let's just go to Spiritus and basically ask if we can use his phone. The others are momentarily stunned by the idea before agreeing it's worth a try. Hilariously, it works.
Even better is when the awkward situation of the heroes casually bunched their mortal enemies comes to Materia's attention:
Shantotto: Don't waste the last drops of our precious time gawking, the situation demands we begin talking! Warrior of Light:(to Noctis, sheepishly) Gawking? Was I gawking? Noctis: What're you asking me for?
Kefka dodges a Pyroclasm from Vaan and taunts him... leaving him wide-open for Tidus to Jecht-Shot the flame back at him. He turns around and has a gloriousOh, Crap! on his face before we see the explosion over Exdeath's shoulder.
Golbez is levitating rocks to throw at Squall as he is fighting Jecht. Bartz lands on a higher one and throws a spear at him. Golbez... just sort of puts his hand in the way, like "No", and gets the rocks going.
Unlike the previous games, all pre-fight quotes are generic; no specific lines depending on enemy, and nothing different in story mode. Also unlike the previous games, the enemy leader and the player's character always speak, in that order. This can lead to some astonishingly inappropriate exchanges that sound like the opposing teams are encouraging one another.
To a lesser extent, most pre-fight and post-fight quotes assume the speaking character is on a team, facing another team - and in fairness, they always are in Story Mode, Gauntlet Mode, and Team Ranked Matches. Not so much in Sparring Mode, Solo Ranked Matches, or Custom Matches. Terra telling nobody that they have to "fight as one", or Onion asking how to get rid of "these wimps" against a single enemy can be funny. Lightning telling off the enemy that they're nothing if they "don't work as a team" when she is alone is amusing. Vaan asking how to "get their goat" when facing the Cloudof Darkness is hilarious.
Then there's the way some characters can be described unintentionally by their allies. Such as Shantotto, teamed up with Squall and Ace, saying "It'd be prudent to treat my teammates as students".
Most people are surprised by the final boss's very immediate move at the start of your fight with it. Because yes, even in this game, Shinryu opens up with Tidal Wave.
New to the game is a chat function, activated with a press of the D-pad and a timely tap of a face button, to prompt a line spoken by the character. Default lines, and others that are useful in battle, include such things as "I'll destroy the summoning core", "I'm aiming at that enemy [identified by number in the text]", and "Sorry I messed up". Optional lines, purchasable in the shop, include some of the most hilarious quotes from each members' original game, as well as certain quotes pulled entirely out of context, all fully-voiced. These can be assigned to be spoken in battle at any input, which means that any member of the party can make an absolutely inappropriate comment such as...
Garland: No one touches my princess!
Firion: Hey, it's free! What an offer!
Onion Knight: Well... Shall I just sleep in the princess' bed?
Ramza: Sheathe your blade, and let us talk! * This is unfortunately impossible, as Dissidia is a fighting game and Ramza makes no appearance in the main story (i.e. the part where talking instead of fighting could work).
Ace: Yeah, I like chocobos. Is there something wrong with that?