Factual error: At one point, Newt Scamander says he found one of his beasts in Equatorial Guinea. That country only got that name after independence from Spain in 1968; back in 1926, when the movie is set, the place was known as Spanish Guinea.
Character mistake: In her speech outside the bank, Mary Lou Barebone mentions "the wireless" as one of the wonders of modern technology, which is the British term. A New Yorker in 1926 would more likely use the Americanism "radio."
Character mistake: Toward the end of the movie, the behavior of Newt is totally out of character - not just for him, but for any sane person. Newt tries to save Credence, and he has just gotten him to stop attacking and actually listening - when the other wizards blast what Newt knows to be essentially a traumatized and abused child to kingdom come. But Newt isn't the least bit distressed or mad, he just shrugs it off in a "shit happens, life goes on" kind of manner, catches Grindelwald for them and helps them save the world, happy as a flea and not giving a damn about the recent death of Credence. Only a complete and utter sociopath would be totally unmoved by the death of a child they were trying to save. A sane, feeling person (like Newt) should be at the president's throat spells blazing, cursing her and the whole American Wizard-hood to the nether regions of hell and not giving a blue damn whether or not they are exposed and in trouble, and whether or not Grindelwald goes free, because in that moment, in his eyes they would be hardly better than him. It is even more out of character since Newt is until that moment depicted as extremely protective of his creatures.