Captain Wentworth is seen wearing his naval uniform. As a captain, he is missing both epaulets and the golden chevrons on his collar. Also, naval regulations of the time would have dictated him to be clean shaven, which also would have applied to his brother the Admiral.
In the final scene when Anne is hugging the Captain, you can clearly see the back clasp of her necklace. It is shown multiple times to be a spring ring clasp, which is very common today, but wasn't invented until the early 1900s, some 80 years after the time period for this film.
Mary writes a letter that says she is ill and uses the sad face of two dots and a down-turned mouth. The happy/sad face doodle was first used in the mid-20th century and not the early 19th century.
The look and words Mr. Elliot has with Captain Wentworth during their first encounter on top of the cliffs would had been considered quite an insult for the time period. Common courtesy would have dictated an immediate apology, followed by a written one. Otherwise, such an exchange would have sparked a duel..