Spock finally fully accepts and admits to feelings he was terribly ashamed of in The Original Series. What exactly those feelings areis up to interpretation, but it's heartwarming either way. Bear in mind that they've both spent the whole movie explicitly Not Mentioning the fact that Spock abandoned the ship after the end of the series to return to Vulcan, because he has rejected all traces of emotion for pure logic, and appears to have willfully forgotten every feeling he ever had for his friends. But when he wakes up...
Spock: V'ger has knowledge that spans this universe. And yet, with all its pure logic, V'ger is barren. Cold. No mystery... no beauty... I should've known... [trails off] Kirk:[leaning over him] Known? Known what? Spock, what should you have known? Spock:[taking Kirk's hand] Jim, thissimple feeling is beyond V'ger's comprehension.
Even more so, because in Vulcan culture, holding hands is considered a very intimate gesture, like a firm hug...or perhaps a kiss.
"I want this, Jim! As much as you wanted the Enterprise, I want THIS."
"We can have you back on Vulcan in four days, Mr. Spock." To which the reply was: "Unnecessary Mr. Scott. My task on Vulcan is completed."
Kirk's reunion with Scotty, and their shuttlepod tour of the refit Enterprise. Did it last too long? Maybe, but the look in Kirk's eyes is almost like he's back with an Old Flame again. And after the tour, Kirk simply says "Thank you, Mr. Scott." once they dock. The latter clip was used in honor of James Doohan during the 2006 Academy Awards' "In Memoriam" montage.
This scene was likely meant to be a reward for the series' long-term fans, who were seeing the Enterprise again for the first time in ten years.
When Bones comes aboard, he complains bitterly about his Mandatory Unretirement. But all Kirk has to say is "I need your help," and Bones drops the act and shakes Kirk's hand. Then he storms out of the transporter room complaining about the ship's new design. Kirk can't help grinning: in spite of the seriousness of what's going on, Bones is delighted to be back on the Enterprise.
A subtle one in the final scene: when reporting on casualties, Kirk respectfully refers to Decker as Captain rather than Commander.