There's the question of whether Moore's reallyTruer to the Text than most other adaptations, or whether he's really just pushing for the darkest possible depictionsfor his private enjoyment. Particular sore spots include: Mina Murray being a divorced woman when she was Happily Married to Jonathan Harker in the original novel, Mr. Hyde raping the Invisible Man, James Bond as an incompetent misogynist psychopathic traitor instead of being a loyal, competent Professional Killer, and Harry Potter as a whiny, self-pitying, school-shooting chav strung out on anti-depressants who becomes the Antichrist, which is pretty far off from his actual character.
Crack Pairing: Since the series deals with the relationships between various fictional characters, this happens quite a bit. Most visibly with Quartermain and Murray, but it happens with minor characters as well. Frankenstein's monster and his wife Olympia from Tales of Hoffman come to mind.
Creator Provincialism: A frequent criticism of the later volumes. For a series that's ostensibly a tribute to the history of fiction, it can strike some readers as a bit strange that most of the coded references in 2009 are to British pop culture, in spite of the growing influence of American and Japanese pop culture in the 21st century.
Genius Bonus: If you got every single reference in this series without help... you need to make a lot more pages here at TV Tropes.
He Panned It, Now He Sucks: A big part of the reaction towards Century: 2009 comes from the fact that a big part of the last leg of the story boiled down to a mean-spirited hatchet-job directed at Harry Potter. Whether fans' reactions were just this trope in action, or whether it was legitimately poorly-done and damaged the work from a literary standpoint is up for debate.
Moore's Grand Finale for Century: 2009 involves an epic face-off between Harry Potter and Mary Poppins. Just a few months after he wrote that scene (and almost exactly a month after the comic hit the stands) a battle between Voldemort and a swarm of Mary Poppinses turned out to be part of the Opening Ceremony of the 2012 Olympic Games.
Among many other tidbits, Century: 2009 manages to tie James Bond and The Avengers together into one universe with the revelation that Judi Dench's M in the later Bond films is actually an aging Emma Peel. Though we never get to find out M's true identity in the films, Skyfall actually did turn out to include a brief moment where Kincade, Bond's old groundskeeper, addresses her as "Emma" (presumably because he misheard "M" as "Em").
The final scene of the comic is of Quatermain's grave in Africa, just like the movie.
Similarly, though the movie had "Fantom" (a villain loosely based on Erik from The Phantom of the Opera and Fantomas) the comics did finally incorporate Phantom of the Opera into the plot of The Black Dossier. According to one of the supplementary stories, the League had their final face-off with France's "Les Hommes Mystérieux" at the Paris Opera, where they tried to stop their plot to plant explosives in the Phantom's old lair. The other half, Fantomas, being one of the French team members.
Alan Moore has long been well-known for practicing ceremonial magic and being an avid student of occultism and the mystic arts, and he (in)famously claimed in 2003 that he worships Glycon, a Roman snake god that was once the center of an ancient pagan cult. In 2011, he attracted a bit of controversy for portraying Harry Potter as a thoroughly unsympathetic Antichrist figure who's also supposedly the epitome of everything wrong with the 21st century. In other words: Moore is an occultist who talks to snakes and has an intense personal hatred of Harry Potter. Voldemort? Is that you...?
Les Yay: Mina has no use for Orlando when he's a male.
Narm: Allan's death. To elaborate: he gets electrocuted by lightning coming from Harry Potter's dick.
YOU ARE THE SHIT OF THE WORLD! I KILL YOU NOW!
Nightmare Fuel: The dead bodies fused to the remains of the train in Century 2009, to say nothing of the flashback panels to the events of the massacre that caused that.
No Export for You: In Canada, at least, you can't buy The Black Dossier without online ordering.
Offscreen Moment of Awesome: We never get to see the full exploits of the Second League of the Extraordinary Gentlemen, and their very-much-indeed awesome-sounding encounter with Les Hommes Mysterieux is only described in text on the Black Dossier. Also sideway referenced in text are the missions of Prospero's Men, The Third League of the Extraordinary Gentlemen, Der Zwielicht-helden and Les Hommes Mysterieux themselves.
We also see far too little of the League of the 1780s, featuring Lemuel Gulliver, the Scarlet Pimpernel and wife, the Scarecrow, Fanny Hill, and Natty Bumpo. Most of what we do see when they appear is when they've largely retired from adventuring and are touring the world indulging their more hedonistic tendencies.
Many who claimed that the portrayal of the Antichrist and Harry Potter was unfaithful to the source material. The Antichrist's personality in Century:2009 as a paranoid teenager confused about his destiny and confused about being a Pinball Protagonist by his mentor is more or less consistent to the Wangst-y arc of the character in Book 5 (which was pointedly excised from the films to make the character sympathetic to the audience, i.e. proving Moore's point), as is the criticism about his obsession of fame and repeatedly noting how famous he is, much of which echoes complaints by Ensemble Darkhorse characters in the books. Likewise the portrayal of Ron and Hermione with Ron bravely standing to try and talk sense to Harry while Hermione cries for her mother is an exaggerated inversion of the dynamic in the movies where Hermione is portrayed as an Action Girl with Ron's best moments given to her.
Schedule Slip: A regular enough occurrence that there's actually a backup strip in the v2 trade about it.
People who disliked Harry Potter or who liked it but felt it was overrated in esteem and especially found the title character less interesting than the supporting cast enjoyed Moore's takedown of it in Century Vol 3. These fans also point out that Moore's basic satirical message, i.e. a Character Exaggeration of his Idiot Hero tendencies and an attack on the stories overall "trust-fund orphan" narrative of entitled heroism and luck-driven victories is in fact completely accurate and moreover echoed criticisms of the book made by its own fans and by Severus Snape within the stories. They note that Snape is the only HP character who is treated positively by Moore.
The same applies for people who enjoyed the trolling of James Bond, even by Bond fans who felt the character was so overexposed they found this revisionist version entertaining. The fact that Jimmy is so hilariously bad at his job and a bungling wimp who can barely get laid makes him less of a Take That! and more of a dark Deconstructive Parody for Bond fans.
The announcement of the Century trilogy initially had fans buzzing because they thought they'd finally get to see the original graphic novel's premise applied to 20th century fiction. And they did... except, instead of creating a new team of champions for a new era of fiction, Moore just made the two remaining members of the original Five-Man Band immortal, and added one consistent new member (Orlando) who quickly devolved into a Creator's Pet. By 2009, Mina and Allan have mentally aged so much that they barely even resemble their literary counterparts (which kind of kills what made the series fascinating in the first place) leaving behind little more than ultra-obscure background references.
A bit meta but almost all of Alan Moore's choices for Century:2009 were recycled cliches about millennials. Millennials are overly reliant on pharmaceuticals, have no culture, aren't involved in politics or society, the list goes on. For all of Moore's supposed counter-culture tendencies, it's very easy to picture him complaining about today's big civil rights movements in the same way Louise Mensch might. How bad is it? The closest Moore comes to approaching what today's generation has to deal with is observing that row after row of houses are empty but quickly back peddles from this and tries to frame it as being the fault of Millennials! That the entire Harry Potter plot line boils down to a Dark!Harry Manipulative!Dumbledore fanfiction is really just the least of Moore's poor choices.
Others felt that the universe of the book continuing to be doggedly Anglocentric in its depicted references—in spite of the growing influence of American and Japanese pop culture—was also far too provincial in scope.
Values Dissonance: The comic deliberately fakes this trope to create aesops such as "The Chinese are brilliant, but evil". Which we would like to stress is a verbatim quote.
What Do You Mean, It Wasn't Made on Drugs?: The Beatnik novella from the Black Dossier reads like this, which, given the source material, isn't surprising. If one takes the time to actually decipher the text, the plot seems to involve Fu Manchu and Professor Moriarty's descendants (Dean Moriarty and Doctor Sax, respectively) continuing a family feud by unleashing an ancient Aztec linguistic virus made from centipedes. Oh and the virus actually turns out to be Lovecraftian Eldritch Abominations.
Broken Base: In a connection to the example for the comic the film gets this a lot too. The film made a lot of changes to the point of being In-Name-Only, this clearly irritated fans who like the comic. But given there is an entry of this trope there too, there were a lot of people who found the comic disappointing (or became disappointing). For this side some of the movie's decisions are called improvements to the comic. Debates about this still spring up to this day on most sites talking about the movie or comics.
The movie's Evil Plan involves a mysterious bad guy (who's eventually revealed to be Professor Moriarty) trying to start World War I a few decades early. Sherlock Holmes: A Game of Shadows, which came out almost a decade later, was about the same thing. In this film, Moriarty even references the Reichenbach falls as where "that man died." Perhaps he got some plastic surgery, and tried to start his Evil Plan all over again, but went more ambitious by using the League?
Richard Roxburgh would also go on to play Dracula in Van Helsing a year later, which also featured a Mr. Hyde who was depicted as a HulkExpy.
Nemo: A man with an unusual beard, untold riches, and access to advanced technology that no one else can duplicate.
Quartermain: A legendary old hero in an era that is not his own, who lost someone close to him while working for his government.
Mina: A beautiful red-haired woman with a traumatic past who dresses largely in black and is much more dangerous than she appears.
Jekyll: A mild-mannered Doctor who, at times, transforms into his large, super-strong and ferocious alter ego.
An attack on the heroes' cool transport by the pretty boy bad guy and his inside knowledge, and he's working for an even more dangerous foe.
And they're all working at the behest of a mysterious government figure. The only ones that don't match are Thor and Hawkeye note unless you include Quartermain, who is a crack shot, or reformed thief Skinner (assuming MCU Hawkeye shares that part of the comic version's past), but other than that, one almost expects Quartermain to yell "League, Assemble!"