For me, I think the most memorable and enjoyable moments for Fallout 4 were the moments when they really went off the rails to do something weird and fun in the game. The stand-out quests for the main game were "The Siilver Shroud" and "The Last Voyage of the USS Constitution."
The Silver Shroud quest is special because of the fact Goodneighbor actually feels it needs you. One of the biggest flaws of Fallout 4 is that in Fallout 3, you were desperately needed and the Capital Wasteland felt roughly between Mad Max and a Radiation themed level of Hell.
The Savior of the Wastelands was needed to give the people a fighting chance or finish them off. In Fallout 4, had the Sole Survivor never woken up, eventually the Brotherhood of Steel would have cottoned to the fact the Institute is under C.I.T. and blown it up with Liberty Prime. You, literally, didn't have to be there to get one of the major faction endings.
Goodneighbor, however, is a place which needs the protagonist and the fact you go on your killing spree as the Shadow actually contributes to making it a better place. If Goodneigbor had been larger, I think it would have been felt more but still I loved being the Shroud.
The Last Voyage of the U.S.S. Constitution was also extremely fun not just because of the silly malfunctioning robots but because it's also about idealism. The Scavengers just want to destroy a piece of history and you have to choose between thinking machines and flesh and blood humans. It actually articulates the Synth question much better than the game and more enjoyably. The Synths are, in their own way, "too human" for the question to have relevance.
I think the only moment which felt as awesome as Fallout 3's entering into the Capital Wasteland from Vault 101 was when you entered the Glowing Sea. Sadly, that lost a lot of its feeling when you returned and discovered it wasn't quite so terrible as they depicted it but it was still one of the few places which felt like it had grandeur and a sense of horror. It was, again, akin to visiting a radioactive hell.
The arrival of the Prydwen is a great moment of Fallout. The Brotherhood of Steel at its height for the first time since Fallout 1.
The descent into the Institute was another moment which felt amazing as they managed to really create an evocative aesthetic for it. It's a beautiful place which is clean, children live in safety, and there's even fountains. However, theres a great visual metaphor as the further you go from the center the worse it looks and you soon find it is an awful place under the surface.
I think the Kellog battle is one of the few conflicts in the game which felt like it was with a decently written character and reminded me of the conflict with Benny in New Vegas. You have a genuine reason to hate this person and the emotion the Sole Survivor expresses is more than most of the rest of the game. Killing Kellog actually felt like it should have been much further into the game because he was a character who could have been mined for much more pathos, I think. I also think it needed a Dishonord, "Spare Kellog" option.
The Companions were overall great and I loved Piper, Danse, Nick Valentine, Cait, and even Preston until Sarge was dealt with. After the Fort was retaken, he became annoying by virtue of his constant unskippable radiant quests. Still, I've got to say the personalities of the characters were great and approached Bioware's ability to make likable companions.