As Flynn enters the barber shop you see a man playing guitar. The music that you hear is a basic blues rhythm that is played with a (metal or glass) slide on the left hand. However you clearly see that the man does not have a slide on his fingers and furthermore his left hand rarely moves. It is impossible to get the tones that you hear without a slide and without more frequent hand movements with both the left and right hand.
Aspen trees are native to the northern parts of North America and the western mountains. They require cool summers (not found in New Orleans) to thrive. It is unlikely one would be growing in the swamps of Louisiana.
Datura hallucinations are quite different from what is depicted in the film. Although, seeing someone grow an extra finger, for example, is possible, datura is more along the lines of temporary psychosis. The most common hallucinations would be seeing people who aren't there, hearing voices, having conversations with people who aren't there, and seeing inanimate objects move and float. The character in the film would have never been able to remain so coherent after consuming a fairly large datura tincture. In reality, he would be extremely paranoid, probably running from or grabbing at his hallucinations.
Simone describes a painting of the first Librarian as being from before the First Crusade. The painting is in a style which did not come into use until the Renaissance. It could be a painting from the 15th or 16th centuries which depicts an event from before the First Crusade, but that is not how it is described.
Jean Lafitte, the famous pirate, removed his pirate base from the New Orleans area in 1817. In 1823 he was wounded while on a pirate adventure and died. He was buried at sea. It would therefore not be possible to find either his skeleton or his ship in the swamps around New Orleans.
When Flynn sizes up the slot in the mouth of the Dagger Man, he says it's about three quarters of an inch in diameter, making the nearest comparable modern coin a silver dollar. However, a silver dollar is more like an inch and a half in diameter; a dime would be closer to three quarters of an inch. He probably meant "radius" rather than diameter.
Flynn fends off the henchman with the gold hand, he uses the stove to turn a cushion into gold. He then whacks the henchman and hands off the cushion to the auction house's cashier. A mass of solid gold the size of that cushion would have been too heavy for an average person to lift, much less whap someone upside the head with one hand.
When Flynn and Simone are trapped on Jean Lafitte's ship much is made of the fact that they are sealed inside a room with solid walls and that their only hope of escape is by breaking down the door with a cannon. Yet, when they are setting up the cannon the stern windows of the ship are clearly visible behind them. All they would have to do is smash the glass and crawl out.
The goof items below may give away important plot points.
Near the beginning of the movie, after the auction is complete and Flynn has paid for the Ming vase, he smashes it on the ground in order to retrieve the philosopher's stone hidden inside its base. The stone, as Flynn explains, turns anything it touches into gold. As he retrieves the stone and pockets it, a rival bidder approaches with a bodyguard, determined to forcefully take the stone from Flynn. The bodyguard takes off a glove and reveals a hand of solid gold, explained away by the fact that the he handled the philosopher's stone on a previous occasion without taking the precaution of ensuring he didn't come into direct contact with it. Only one problem with this scenario - if the goon handled it on a previous occasion, how did it end up encased in a priceless Ming vase, hundreds of years old? Flynn had to smash a valuable antique in order to get to the stone. The vase wasn't a fake, it was the genuine article.
Flynn uses the recoil of an unloaded cannon to break down the door in the ship. He loads in powder then stuffs his coat in as wadding. When the cannon goes off it recoils violently. Guns recoil primarily because they are firing a projectile (Newton's third law of motion). Without a projectile, as when firing a blank, there is only a modest amount of recoil, certainly not enough to do what is shown.