According to the books and Pottermore.com (site created by JK Rowling herself), Minerva McGonagall was born in 1935 and first started teaching at Hogwarts in 1956, when she was 21 years old.
Even though the movie is set in 1927, eight years before her birth, she appears as a young professor in Hogwarts.
When Dumbledore and Newt are talking in the bus, Dumbledore's hat is crushed and not crushed on the top depending on the camera direction.
Grindelwald's hair length between the previous movie and this one: only 6 months have elapsed but his hair is now quite long.
When Newt and Tina are caught infiltrating the French Ministry of Magic, Newt can clearly be heard saying that the cats are 'familiars', despite the fact that on Pottermore.com, screenwriter and Harry Potter author J.K. Rowling clearly states that familiars "do not exist within the world of Harry Potter."
However, Newt actually states that the cats are "spirit familiars", which may be something very different from the traditional concept of familiars that Rowling stated do not exist in the Wizarding World.
Nagini was introduced in a show as an animal from Indonesia, even though Indonesia became independent only in 1945. In 1927 it was generally known as East Indies. However, the name Indonesia has actually been known since the 18th century, especially in academic circles. In late 19th century it was popularized in Adolf Bastian's book, and by 1920's it was known well enough (at least outside of the Netherlands) to explain its use in the context of this movie.
Grindelwald, young and present, is shown to have a pronounced eye pigmentation defect that wasn't there in his portrayals at different parts of Harry Potter and the Deathly Hallows: Part 1 (2010). While it could be argued that this stems from an incident that took place somewhere in between these films, rather than a genetic phenomenon since birth, that still doesn't explain why it's not there in his older age. There remains the possibility that this matter may be explained in Fantastic Beasts 3, 4, or 5.
In this film, we see a young Dumbledore teaching Defense Against the Dark Arts in 1927 at Hogwarts. The Harry Potter book series only ever mentions him as being the Transfiguration teacher. Dumbledore having taught Transfigurations does not preclude him from also having taught DADA at one or more points during his tenure. He had a very long life and was known for being an expert in multiple fields.
Dumbledore says that he sent Newt to New York. In Fantastic Beasts and Where to Find Them (2016), Newt says the reason he came to America was to send Thunderbird home to Arizona. It is implied that the Thunderbird mission was a cover for Newt's actual mission for Dumbledore. It's certainly possible that both missions were considered to be of equal importance to Newt.
Obvious aging mask on Brontis Jodorowsky's face particularly visible around the eyes.
If the water entered the coach so rapidly when riding on/in the river, it would be able to drain as fast and not be stuck and drown the people inside.
In the London street scenes, the London Underground symbol on station entrance signs has a solid red disc in the centre. This is incorrect for 1927 when the film is set: Edward Johnson redesigned the original 1908 solid-disc symbol in 1919 and changed it to the "roundel" - a red ring with a white centre - which was introduced in the early 1920s and is still used today. The blue horizontal bar remained unchanged. However it is plausible that some old signs had not yet been changed to the new London Underground house style.
One of the extras, a Frenchman whispering to his girlfriend at a café asks her "Tu sais quoi?" which is "You know what?" in English; This has been a common Anglicism for several decades but was not used 100 years ago.
The couple sitting at the café and overheard by Kowalski say "Tu sais quoi? - Dis-moi - J'te trouve très belle" (You know what? - Tell me - I think you are very pretty) but their lips don't match their lines. (56'05")
Ministry wizards arrive at Hogwarts by Apparition, but it was established in the Harry Potter series that this is impossible within the Hogwarts grounds as there are enchantments in place preventing it, which can only be lifted by the Headmaster. Judging by Professor McGonagall saying that the officials had 'no right', their visit was unexpected and so the Headmaster wouldn't have lifted the enchantments at that point.
Professor McGonagall is shown as already teaching at Hogwarts during Newt's time as a student. This is a completely unexplained anomaly as it changes the established canon from the original Harry Potter series, in which McGonagall was born in 1935 and did not begin teaching until 1956.
According to information JK Rowling included on the Pottermore website, the Defense Against the Dark Arts Professor at Hogwarts during Newt's school years was Galatea Merrythought, who taught from 1895-1945. Dumbledore did not become a teacher himself until the 1920's, and throughout the original book series was always referred to as the "Transfiguration teacher" before becoming Headmaster.
Seraphina Picquery lets Grindelwald's wand be carried along with him in the coach. She just said they needed to be extra cautious as Grindelwald could escape and is dangerous. Certainly an easy way to get his wand back.
Grindelwald's mustache keeps changing in length, especially in the second part of the movie. Sometimes it's shaven above his upper lip and sometimes it hangs 'on' his lip.
Maja Bloom, who plays the character Carrow, is known to be multilingual (English, French, German, Polish and Spanish) but she interprets a French character and speaks French with a moderately heavy English accent (then fakes a French accent when speaking English).
Johnny Depp gives Grindelwald a British accent, but Grindelwald was born and raised in the Austro-Hungarian Empire.