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.
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: 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
Continuity mistake: In this movie, Emma Frost is seen as an adult woman [roughly around the same age as Professor X and Magneto]. But the character is also in X-Men Origins: Wolverine, set years later when Professor X and Magneto are much older, but in that movie she's younger than she is in this one. Although in Origins she is Kayla Silverfox's sister and her surname is never heard - the character was advertised and created as Emma Frost.
Factual error: When sitting in Xaviers mansion there are cushions on the sofas made from a Sanderson fabric, Dandelion Clocks, this fabric was designed in 2009 but the film is set in 1960.
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.