Continuity mistake: In First Class, Professor X is made handicapped in 1962 at the end of the movie. In the previous X-men movie, he was walking when he visited the girl (Jean) in the mid 70s with Magneto.
Factual error: Erik arrives in Argentina at a place called Villa Gesell, shown to be a town in middle of mountains. Villa Gesell is a beach town on the Atlantic Ocean coast, with no mountains.
Factual error: In the scene where they first pull up to the mutant research facility, they are in a 1972/3 Cadillac when the movie is set in 1962.
Continuity mistake: In a scene where Xavier is pointing a gun at Magneto the distance between the gun and Magneto's head changes between shots.
Continuity mistake: In the original X-Men film, Xavier mentions that he met Magneto when he was 17. In this film, he doesn't meet him until well into his 20's.
Other mistake: The knife used in the Argentina pub scene never has any blood on it, despite being driven into one man's stomach and another man's hand-twice.
Factual error: When the US & Soviet navies attack the mutants on the beach they fire a large volley of surface to surface missiles. This is supposed to be 1962, the Cuban Missile Crisis, and ship launched surface to surface missiles had only just been introduced and weren't very common. But what makes this a technical error is that in the barrage we see RGM-84 Harpoon anti-ship missiles, introduced in 1977, and RGM-109 Tomahawk missiles (could be anti-ship or land attack) developed in the 70s and first entering service in 1983.
Factual error: The aquarium in the lounge area of CIA facility where the young mutants gather has several blood parrot cichlids in it. This is a hybrid ornamental fish developed in Taiwan in the mid 1980s and did not exist in 1962.
Continuity mistake: In the scene where Erik confronts Shaw on the boat, Emma's hairstyle changes from being down to being in a half updo right after Shaw chastises her for harming their own kind.
Factual error: A Westland Bell 47G helicopter with the a registration of G-CHOP is seen to land at the Russian army camp/military headquarters when Charles, Magneto and the X-Men are approaching. The registration denotes it is a British aircraft, so unlikely to be in the middle of Russia, and this particular model was built in 1966 - four years after the Cuban missile crisis
Factual error: In one scene when Sebastian Shaw talks to Emma Frost in the submarine you can see the TV playing in the background, and right next to it is a Revox A-77 reel to reel machine. This movie is set in 1962, but Revox didn't introduce the A-77 until 1967.
Continuity mistake: When Banshee is diving into the water, his "wings" are still attached. The shot angle changes, and his wings have disappeared before hitting the water.