The "death" of the Enterprise D may not be as heart-wrenching as that of the original ship, but when it's out-right stated that she can't be salvaged, that's when it hits you.
What about that slow pull back once the saucer has come to rest? There's a moment of relief at the realization that they made it through, and then it really hits you - the crew survived, but the Enterprise won't be getting back up from that one.
The "final" death of James T. Kirk, Captain of the Enterprise.
"It was the least I could do... for the Captain of the Enterprise. It was fun.... oh, my..."
Picard talking with Troi about his family, who he has just learned had all but one died in a fire. He manages a few words with Troi before he breaks and just starts sobbing over the loss of his nephew Rene, to whom he was exceptionally close. All he's been through in his career and life, and that the man is still capable of showing such emotion makes for an intensely powerful scene. Even worse is his attempt at a Stiff Upper Lip by saying it's all right - and Troi replying, aghast, "Captain! It's not all right!". After so long as the calm, collected Captain of the ship, Troi has to remind him that it's okay to fall apart after such news.
Picard giving a Nexus recreation of Rene a hug.
Picard having to abandon his fantasy. He is so close to just letting himself get sucked into the illusion; here, he has a family again after his real relatives were so cruelly taken from him just days before. Even the illusory Guinan doesn't attempt to stop him- but he knows that to stay would be to condemn millions of lives to death as the Veridian system's sun goes nova, and in the end, he lets his fantasy go for the sake of the many.
Kirk sounds so broken when he accepts that the Nexus isn't real, despite him wanting everything so much. He has lost just as much as Picard in his life, though his losses weren't quite as fresh.
Cat owners identify with Data crying over finding Spot.
After seeing the Enterprise-B axillary control room Kirk was in is now just a massive hull breach:
Chekov: My God! Was anyone in here? Scotty:(solemnly) Aye.
Behind the scenes, imagine how devastated Spock and McCoy were when they got the news of their best friend's (supposed) death.
If you read the novelisation, you don't have to use your imagination. It's as devastating as you'd expect.
It's worse. Had Kelley and Nimoy been in the roles intended for them, imagine this scenario.
McCoy: My God! Was anyone in here? Spock: Yes.
William Shatner himself has said that this following line was the hardest he ever had to deliver:
Who am I to argue with the Captain of the Enterprise?
What about Scotty's report on how many of the refugees he managed to transport off the failing refugee ship?
"I got 47..... out of 150."
Sulu's daughter reading off the damage report on Engineering, hesitating over "... and Deck 15" where Kirk was, and then just sitting there shocked and horrified that the man her father's been telling her incredible stories about since she was a baby is gone. And somebody's going to have to tell her Dad about it.
Soran: No, no! You don't understand! I have to go back! I have to go BACK!
Many people have questioned how a Starfleet icon like Kirk could ever seriously consider staying in the Nexus and not trying to save the day? He explains, he's spent his whole life saving the galaxy, and all it got him....was an empty house. There are several people who can relate to feeling they missed out on things because of devotion to career.
It speaks volumes that even with above, when he finally stops to think about it, he totally goes with Picard. James Tiberius Kirk will never EVER not do his job. Not even in (kinda sorta) death.
Not precisely. He goes because it's not real, not because it's not desired.
Another reason is a subtly sad moment when Jim is shocked Sulu had a daughter, someone who was climbing the Federation ranks just as hard as he was. It was his shocked realization that he had no one in his life at all. No Carol. No Antonia. No David. No one except his shipmates (who he's now mostly separated from), and that isn't quite enough. No wonder he'd jump at being able to fill "the empty house".
The novelization has one that's sadly missing from the movie; Captain Sulu and the crew of the Excelsior are running an evacuation drill, when a grieving Chekov calls in and interrupts (this is after the Enterprise-B escapes the Nexus). Sulu immediately thinks something has happened to his daughter, but when Chekov reassures him that she's fine, Sulu starts to wonder why else would Chekov be calling — and then it hits him...Kirk.