Tied in with Assassin's Creed: Unity, the ending has Shay talk about how the Templars need to start their own revolution. Which means that in a backhanded way, the Assassins did start the French Revolution.
The entirety of the first sequence, Shay is bringing up objections to the Assassins and their methods. He dislikes the way they support the French against the English while working against the French in Haiti, he dislikes the way they insist on blind obedience despite preaching freedom, he dislikes how the war against the Templars is a Forever War with no possibility of negotiation, and feels unclean after murdering a pair of helpless old men (which the Assassins all but high-five themselves for after). It makes his subsequent defection much easier to understand.
This also underscores the points earlier games made about why Templars find recruiting relatively easy and the Assassins can never keep their numbers big for long. Complex problems with almost exclusively bad solutions aren't something a person can stomach for long. The Templars may be, well, Templars, but they offer clear answers which makes swallowing all the obviously terrible things you have to do easier.
By the end of the game, Shay has gone from a seemingly benevolent, albeit rash and angst-ridden, young man who wants to protect innocent life to a typical Templar bastard willing to spark a bloody revolution just to undo some of the damage Connor did to the Templar Order. One might simply chalk this up to typical moral decay. However, this is arguably foreshadowed as early as Shay's assassination of slaveowner Lawrence Washington. Despite Lawrence being an undeniably detestable person who deserves punishment (and previousprotagonistsbeing opposed to chattel slavery), Shay feels guilty over taking his life. This "both-sides" approach to morality ("Sure, the Templars are bad, but the Assassins are worse because they kill old people!") shows that he has always had a short-sighted, naive perspective which refuses to see nuance ("Killing is usually wrong, but killing a slaveowner - even an old one - is justifiable.")
Edward Kenway and Shay are Foils to one another. Edward was a Welsh pirate who, eventually, decided to become more responsible by joining the Assassins. He started out working for the Templars, however briefly, then spent many years hunting them down for his own selfish purposes. He sought the Observatory for the wealth and power it would give him. Shay is a New York City Irishman turned privateer who had a criminal background but no real aspirations to wealth or respect. He ends up joining the Templars out of a desire to do good and keep the Pieces of Eden from the hands of the Assassins. Gradually, Shay becomes more and more brutal to make sure his goal is reached.
The Artistic License History entry talks about how it is anachronistic for Shay to allow the singing of sea shanties as a privateer. However, Shay has always valued freedom (at least during much of Rogue) and has a soft spot for working class men, which his crew is. He probably turned a blind eye to their singing as long as other aspects of discipline are met, and the crew gladly obliged.
Juhani Otso Berg seems peculiarly interested in Shay Cormac's life. If you read his background, though, you'll note Juhani has a lot of Pet the Dog moments despite being a Templar. Of course he'd be interested in someone who joined the Templars not for power and wealth but to protect the innocent. They're also both soldiers and killers for the Templars who worked their way up through the ranks to a leadership position.
Juhani killing the Templar agent who scares his daughter, at first, seems like he is just being a Knight Templar Parent. However, the action also serves as an excellent test of just how serious Vidic is about his offer.
Colonel George Monro seems like a really nice guy for a Templar. However, if you look at his actions, they take on a much darker cast. Colonel Monro fishes Shay out of the ocean, places him with a caring family, and then offers to help him oppose New York City's gangs after arriving too late to back him up. It's awfully convenient, though, that all of these events happen in such quick succession. Furthermore, Colonel Monro never lies to Shay precisely but conceals the fact the gangs are allied to the Assassins, that Shay is killing Assassins, and that he is a Templar. In retrospect, it's very easy to see this as a Manipulative Bastard attempting to make use of a pet Assassin until he either proves his loyalty or outlives his usefulness. May double as Fridge Horror since it works.
Shay's outfit (and the Templar outfit from Black Flag it's based off of) is referred to as a Templar uniform, which seems odd considering it's the first we've seen. But if you look at it as a uniform in the same way the Assassin outfits are, it makes sense. Gist and Monro both wear mantles and coats, and many previous Templars wear similar style clothing. Templars have mantles like Assassins have hoods!
Many people have rightfully called out that Rogue is more or less similar to Black Flag because of its presentation and gameplay. However, you're using the same Animus provided by Abstergo, so it's justified in-universe.
This marks the first time in the series when a major city - New York - reappears in a later instance. However, New York in Rogue is way larger and more detailed than it was Assassin's Creed III. There was the Great Fire which withered the place somewhat, but that does not explain how the entire landmass the city sits on seems to have shrunk down between the time of Shay's forays and when Connor set foot in it. The brilliance sets in when one realizes that the cities in the series are not accurate representations of how they were at certain points in history - but how the protagonists remember them. Connor was only a sporadic visitor to the place, never staying for more than a few days at a time, whereas Shay was born and grew up in it, so no wonder he recalls its every nook and cranny in a way more intricate manner than Connor ever did.
Not to mention the Animus in III was made by Assassins on the move, with a serious Celestial Deadline, whereas the one in Rogue is made by Abstergo, with a (probably) infinite budget, no immediate rush to make it, and many more hands operating it, rather than just Rebecca and Shawn. Small wonder then that it's got more room for detail.
Juhani's sending the entire memory sequence to the Assassin seems like a strange thing to do, especially given the Assassin's reaction. Then you have to remember the game is about an Assassin deciding that the Creed is complete crap. Juhani is making a psychological strike against the Assassins where it will hurt the most: their moral certainty.
If that's what he did he's even more delusional than the average Templar. It's the Templar Order which is convinced of its absolute moral certainty. Hence why they feel it's their duty to control the entire world.
The main stronghold of the Assassins in New York, where you find Hope late in the game, is a large walled manor surrounded by gardens and reflecting pools. In other words, it's an American equivalent to Masyaf!
Shay and Connor essentially cancel each others' life's work out. All of Shay's efforts to stop the Colonial Assassins is undone by Connor's efforts to revive them. However, all of Connor's efforts to prevent the Templars from gaining a foothold in America are pointless because Shay is arguably far more dangerous than Haytham ever was. Worse, with Achilles dead, Connor doesn't even know Shay exists. It gets worse as Shay knows where Connor and the Colonial Assassins livenote and it's not confirmed whether the "Connor" that writes to Eseosa in 1804 offering training at the Davenport Homestead — nominally confirming his survival — is even the real Connor.
Shay and Connor are mirror images of each other in many ways. Connor is someone tirelessly devoted to destroying the Templars because of moral reasons while Shay is equally devoted to destroying the Assassins for the same. Both of them have personal vendettas but don't let them influence each others' goals but are chiefly devoted to protecting the innocent. Yet, because of their organizations' feud, they'll probably end up trying to kill the other.
Achilles being a broken and defeated shell of a man is something you did as a player of Assassin's Creed: Rogue.
Several NY gangs of this period acted as liaisons for runaway slaves by pointing them towards ship captains who would take them aboard and sometimes even helping them further north towards Canada. Black gangs (naturally) took this a step further by attacking slave catchers and their vessels that typically sat in the city's harbor. And you wiped those gangs out.
For all the benevolent facade put on by the Templars, the fact is its members include Lawrence Washington, a slaveowner as well as James Wardrop, an ethnic cleanser responsible for many massacres against Native Tribes. Both of them are described by Christopher Gist as "good men". By sheer Moral Luck, Shay allies with them, but it's hard to escape the sense on playing the game, that by taking out the multicultural Assassins, Shay is siding with a bunch of white supremacists.
This leads to the worse bit of fridge horror to come out of Rogue; so long as you give someone a sad death scene people will forget every horrific thing they've done in life and pretend the philosophy that spawned their ruthlessness is perfectly valid.
Despite Lawrence Washington explicitly telling his fellow Templars he doesn't want his brother involved in Templar business, by the time of Assassin's Creed 3, none of his associates are around anymore, and the new grandmaster, Haytham Kenway, plots George's death. Shay's work as both an Assassin and Templar laid the groundwork for nearly killing a man innocent of the two sides' fued.