In X2, Hank McCoy appears in human form. In First Class, he goes through his transformation.
In X2, Sebastian Shaw is speaking with McCoy on television. In First Class, he is killed by Magneto.
In X2, Xavier says he worked with Magneto to build Cerebro, while in First Class Hank McCoy builds it.
In X-Men, Xavier doesn't know why he can't find Magneto using Cerebro, but in First Class Xavier witnesses the effects of the helmet Magneto wears, though the helmets are different designs.
In The Last Stand and Wolverine, Xavier is bald and walking, whereas at the end of First Class Xavier is wheelchair bound.
In The Last Stand, Moira MacTaggert is portrayed as being a British Doctor while in First Class she is an American CIA agent.
Storm and Cyclops appear as children when Charles uses Cerebro for the first time in 1962. Both mutants are children, which would place their adult selves in the first film at older than they actually were.
In the opening of The Last Stand, Eric and Xavier are working together to recruit Jean Grey in the 1980's or 1990's.
Emma Frost appears as a teenager in Wolverine, in the 1980s. However in this film set two decades earlier she is a full grown adult.
In the first X-Men movie, Professor X tells Logan that he met Erik Lehnsherr when he was 17 years old. Although there is a difference of 18 years between 1944 and 1962, it is clear that Charles and Erik were born before 1944 as they were both depicted as young boys at that time in First Class. According to First Class, they met for the first time in their mid-20's in 1962.