"Book/Comic Conan" vs. "Movie Conan". Or rather fans of the latter who use it as their baseline for everything Conan vs. those who are at least aware of the stories. The Shallow Parodies more about Arnold than Conan don't help.
Robert E. Howard vs. other Conan authors.
In particular, the influence of L. Sprague de Camp on the Conan mythos. True-blue Howard fans tend to dislike de Camp's writing and and interpretation of the character, often condescending to Howard's; some trace everything they dislike about how Conan is known in pop culture to de Camp's influence. On the other hand, if it weren't for de Camp and Lin Carter publishing the Ace paperback version of the stories, the character and Howard himself might have disappeared down the memory hole.
Brian Wood's Queen of the Black Coast adaptation is also considered one, since it not only ignores Robert E Howard's tales but the stories that came before (portraying Conan as unseasoned when he had already gained experience); that it goes out of it's way to AVOID having pirate adventures is seen as a problem too.
Ensemble Darkhorse: Thoth-Amon. While a minor character who never met Conan face to face, possessing a number of redeeming traits despite being a villain made him extremely popular, so much so that any adaptions feature him as the Big Bad or an Expy of him.
Fair for Its Day: While many of his female characters were stereotypical cringing females waiting to be rescued Howard also wrote some surprisingly strong female characters: Belit, Valeria and the Devi of Vendhya. His treatment of non-whites is more disjointed and complex. On the one hand is the revoltingly racist "Vale of Lost Women." On the other hand Conan is surprised to find in "Queen of the Black Coast" that his black crewmen, who he had expected to panic and run had fought and taken a toll of the werehyenas. The black guardsman who seeks to kill, and inadvertently frees, Conan in "The Scarlet Citadel" is given a sympathetic treatment. Yes, he wanted Conan dead but for a perfectly acceptable reason - in Conan's pirate days as 'Amra the Lion', Conan had burned his village and killed his brother.
While Howard's most famous Conan villain, Thoth-Amon, was non-white and extremely wicked, he had several noble qualities; he was brave, strong, intelligent, and genuinely cared for the advancement of his people (a quality Conan wouldn't acquire til he took Aquilonia's throne). In "Queen of the Black Coast," the Shemite Bêlit is described as being so mesmerized by the beauty of the treasure she and Conan find that it ultimately leads to her death. This is portrayed less as a Greedy Jew and more as an aesthetic fascination akin to what you might expect from Dwarves. "The Shemite soul finds a bright drunkenness in riches and material splendor, and the sight of this treasure might have shaken the soul of a sated emperor of Shushan."
"Funny Aneurysm" Moment: In the letters page of Conan the Barbarian #267, a reader asks for back issues of the title, and expresses regret that the title was cancelled. The editor points out that a) Marvel don't directly give out back issues and b) the title's clearly still going. That title was cancelled by #275.
Hilarious in Hindsight: The essay "The Hyborian Age" states that the red-headed Vanir conquered Stygia and built the vast empire of Egypt with the early pharaohs being descended from them. This is before the discovery that Ramses II was red-haired.
Ho Yay: The scene in "The Pool of the Black One" where the Black Ones capture a handsome young pirate and force him to dance to a hypnotic flute is described in rather homoerotic terms.
Just Here for Godzilla: Many who bought the original books had no interest in reading them at all. They bought them for the lush, power paintings that served as the book covers. Courtesy of Frank Frazetta himself.
Les Yay: Numerous examples, generally on the part of villainous or 'depraved' characters.
The Valley of the Lost Women features strange women capturing and kissing the female lead in order to paralyse her.
Also a little between the Jenna and Zula in the second movie.
In Red Nails, Valeria assumes that Queen Tascela wants to drug her in order to have her way with her. She actually just want to suck her soul in order to keep her youth. There's also a scene in which Valeria whips the slave girl who tried to drug her.
In Dark Horse's continuity Thoth finally shows that he's irredeemable by callously brushing off the death of his beloved sister in a plague HE Caused.
Narm: At the end of The Scarlet Citadel, Conan decapitates a necromancer. Another sorceror, in the form of an eagle, swoops down, grabs the severed head, and flies off, laughing madly. The decapitated corpse then gets up, and staggers off in (classic, slow zombie-style) persuit. Apparently, this is supposed to be utterly horrifying.
Tearjerker: Surprisingly quite a few. In the Howard stories there's the funeral pyre Conan gives Belit; in the Dark Horse comics there's Conan's failing to save Iasmini and the Aesir from the "Day of Farewell" and Nestor's death.
Values Dissonance: The vast majority of the characters are not meant to be moral even by the standards of the 1920s. However, there are a number of instances - the treatment of women and the veiled horror at "mingled races" among them - that may grate. Howard's plucked-from-history world building method also naturally led to racial stereotypes - just try to find an honest Zingaran, a gentle Pict or an atheist Stygian. But see also Fair for Its Day.