Gozer has been defeated but they see the charred remains of Zuul/Dana. Just check the look on Peter's face when he believes Zuul/Dana is dead. It's the first time we see Peter seriously sad in contrast with the great snarky jerk he was throught the movie.
After that, hope comes and they see movement inside the demonic dog's charred corpse. They break open it, and inside is Dana, dazed but unharmed. The music makes it better because it's a reprise of Dana's Theme and the whole sequence is just perfect.
When their very first call comes in. Janine, who had been largely apathetic up to this point and had made no attempt to hide such, roars "WE GOT ONE!!!" and slams the alarm buzzer. It shows that despite the way she acts, she really does care about the business.
After Venkman successfully makes a date with Dana, he just begins spinning in the Lincoln Center plaza out of joy as the roller skaters dance around him accompained with Bernstein's awesome triumphal background music.
When Venkman makes his speech to the Mayor, especially the part about saving the lives of millions of registered voters. He makes an expression that's a split between genuine happiness and about to cry. It shows he really cares about the city enough to accept any idea that saves it, no matter how damaging or crazy it is.
Well, mainly he's realizing that the Ghostbusters present the means for him to safeguard his re-election.
In and out of canon, the Ghostbusters, using the uplifting energy of the song "Higher and Higher", bring the Statue of Liberty to life and commands her to shatter the armor of hate that Vigo encased the museum in. The sight was so inspiring that it united the normally cynical and hateful New Yorkers, and such was the concentration of their love and hope that it weakened Vigo enough to allow the Ghostbusters to defeat him.
The rallying speech that Ray gives which inspires this plan is also a great moment.
Ray: You know, I just can't believe things have gotten so bad in this city that there's no way back. I mean, sure, it's messy, it's crowded, it's polluted, and there are people who would just as soon step on your face as look at you. But come on! There's gotta be a few sparks of sweet humanity left in this burned-out berg. We just gotta find a way to mobilize it!
I think one of the most heartwarming things in the second movie is watching brave little Louis Tully in a Ghostbuster uniform standing infront of the hate-slime covered museum. You can tell he's really scared at first, but a second later he goes into full on Badass Hero Mode.
Louis: I'm here with you guys!
And after that he fires his Proton Pack's blaster at the shell and it shatters (of course he had a lot of help, but don't tell him that).
Louis: I did it! I'm a GHOSTBUSTER!!!
He may well have done some actual good. Appearing to single-handedly dispel the shell may have boosted the spirits of the watching New Yorkers, allowing them to more easily neutralize Vigo's slime.