The real Henry V had a large scar on the left side of his face, the result of being struck and nearly killed by an arrow at the Battle of Shrewsbury in 1403. The scar is not shown in this film.
Henry V's reign was in the early 1400s, but most of the costuming in the film is from 1600, the time of the plays writing, almost 200 years later. The armor on the other hand is accurate.
In fact, there is no anachronism in the costumes. The story is told from two points of view (one in the 1600s, as a performance in the Globe Theater; the other in the 1400s, as the characters originally lived). Costumes shift on purpose according to the point of view.
When the Archbishop of Canterbury is making his case for war, he recites a list of past French kings and their relations to each other. He gets some of the names, numbers, and relations wrong. This error is taken from William Shakespeare's text.