The ending and the epilogue, Edward sees his daughter for the first time and returns to London. (This scene doubles as a Tear Jerker). In the epilogue, a few years later, Edward watches The Beggar's Opera with Haytham and Jenny. Considering what Edward went through, he deserved this ounce of happiness.
The expression of happiness on Edward's face when he sees his daughter is perhaps the most beautiful moment in the franchise. Edward realized that none of the money and plunder was as great as a treasure as his own child. Also how he slips into Dad mode when he reminds his daughter that the Jackdaw is not a boat but a ship and then becomes the world's coolest Dad by promising Jenny that he'll let her steer the ship when the wind's a little safer.
When Edward sees Jack's skeleton in the Jamaican prison, Edward remarked that as much as Jack wasn't a good friend to him, he still made him laugh and that was enough to make Edward feel sorry to see him like that.
Doubles as a Tear Jerker but the fact that in the end, Edward imagines all the dead pirates at one table including Benjamin Hornigold and Charles Vane, who he was at odds with, the fact that it implies forgiveness is touching, that Edward will remember them at their best despite everything.
Even better is Stede Bonnet at the table as well. Despite his bumbling and incompetence at piracy, and being the last person on Earth for the privateer life, Edward still considers him a good friend.
The fact that Edward never gives up on Caroline throughout the game. It shows how much he still loves her even if she stopped believing in him. One of the first things he does after fixing himself up was write a letter to her. But she was dead by this time anyway and Edward accepts that It's All My Fault why that relationship went sour.
Edward's great redeeming quality that when it comes down to it, he will do right and stand by his mates. His friendship and loyalty to Blackbeard, James Kidd and later, Adewale balances his mostly ruthless, determined edge. And his genuine camaraderie with Stede Bonnet.
When Stede Bonnet gets his own ship, he tells Edward that he cherishes their friendship more than all the plunder and rum in the Caribbean.
It also has a nice moment where Adewale corrects Edward's period appropriate ignorance about Adewale being from Africa (he was born in the Caribbean) and wanting to go back (he doesn't anymore than Edward wants to go to France). It shows Edward is learning to be a more tolerant and enlightened person rather than just being one magically out of the box.
The final missions are moving to, in that we see that Edward slowly but surely change himself. He's less cocksure, more respectful and kinder and even honest to admit that he still has a lot of work left to change himself.
Mary Read's final moments are heartwarming to, for all her frequently expressed disappointment in Edward, she never loses hope that Edward would change, becoming the only person in Edward's life who never lost faith in him, even if she didn't become his enabler.
Also the way, Ah Tabai talks to Edward with a knowing look when the latter admits he was wrong about the Creed, you can sense that the Assassins were all playing Stealth Mentor to Edward, just like they did with Ezio during Assassin's Creed II only they wanted Edward to come to his realization and wisdom on his own and gave him freedom to do so, even showing extraordinary restraint and forgiveness. The fact that the Templars never care to show anything like the same patience or compassion, looking at the false-Duncan Edward in the opening as a thug with no interest shows that at the end of the day, the Assassins give people a fairer shake than the Templars and will practice it too.
Sending a ship to Kenway's Fleet results in a sweet cutscene where the new captain is congratulated by his ecstatic friend. Also the Adorkable expression when Edward puts a Nice Hat on his head.
You will get a bit misty eyed when you see the crew congratulating a black crew member on his ascension. Given they originally wouldn't accept Adewale as captain, this goes to show how the bonds have strengthened on the ship. It also gives the Pirates much Cool People Rebel Against Authority cred, yes we're plundering, murdering assholes but we don't profit by slavery like those jerks in the Spanish and British Empire and we're way better on workers and minority rights than any government at the time.
A darker example is that the Pirate Hunters also contain black crew members fighting alongside each other. It made me realize they were the one group I didn't despise in the Caribbean. It also helped their Not So Different vibes to the pirates they're fighting.
Whenever Edward Kenway cuts the ropes on some of the prisoners and welcomes them onto his crew. Sometimes, admittedly, Edward spoils it by adding a "Join us or die" but most of the time, they look sincerely grateful. It's a sign that Edward doesn't want to kill them all, just wants their loot.
This is Truth in Television - one of the fueling factors of the Age of Piracy is that, in this span of time, being a pirate or privateer was often a much better life for a deckhand than being a Navy recruit. Conditions on military vessels were absolutely inhumane in many cases, and officers had only a low glass ceiling to look forward to as promotions were generally due to nobility or aristocracy rather than merit, experience, or length of service. For your average deck hand, being recruited by pirates was one of the best things that could happen. Also bear in mind that it was not uncommon for the navies of these empires to be largely staffed by unwilling recruits who were pressed into service to escape legal punishment, pay off debts, or in some cases just outright kidnapped and pressed into service.
Edward's many Villainous Rescue of pirates across the game. Edward recruited, quite likely, the majority of his crew from shipwrecks and people executed.
Related to What You Are in the Dark, but the mission where Edward and Black Bart come to the Observatory, Bart mentions that there are native Taino protecting the forest and encourages Edward to Kill 'em All even voicing doubts if Edward has what it takes. The Full Synchronization objectives(Edward's canonical actions) state, to render all Guardians unconscious with bare fists, non-violently incapacitating them. Edward is still a crook but he's not a brainless killer or a Mighty Whitey and this small moment, with no Assassin or Templar to look down their noses on him, gives a sense of his real character, that he's not going to kill a bunch of innocent people who are only protecting sacred land from interlopers.
Cockram and Burgess's final words show a rare moment of the Templars being Not So Different from the Assassins:
Cockram: You done us good, Kenway. Proved yourself a true bravo. And for what?
Burgess: His own bloody pride.
Edward: You stepped in the path of my prize. Not a thing a man should do.
Cockram: A cocksure cully, just like Hornigold said.
Edward: That Templar scab means nothing to me. None of you do.
Burgess: And you're worse for it, Kenway! It were the Templars who took us in when all else went to shit! Not our king. Not our country. The Templars.
Cockram: The Templars is our family. Where's yours?
While they now differ entirely in position, methodology and position, the last words between Grandmaster Torres and Edward has a surprisingly sweet moment towards the end. Torres asks Kenway if he honestly believes that killing him rights a greater wrong then he ever did, and Edward responds that Torres would see mankind safe and sober but sapped of all spirit, so yes, after all this time, he does believe that. The response Torres gives is surprisingly respectful.
Torres: You wear your convictions well. They suit you.
In Freedom Cry every time you rescue a bunch of slaves becomes this.