The telegram to John Groberg uses the actual punctuation mark, the period, but also uses the word "stop." A telegram did not have punctuation marks substituting the word "stop" for the period. It would not have had both as did the telegram in the film.
In the opening shot, the moon is shown in line with the Big Dipper, meaning it would be due north. The moon always appears in the zodiac.
When John Groberg is talking to his girlfriend, Jean, about the moon, he says that the light traveled 93 million miles to reach her, but that is actually the distance from the Sun. Of course, the light originated from the Sun not the moon, so he is correct.
The majority of Tongan characters speak with New Zealand accents throughout the film, although the only white people they have supposedly been in contact with are the American missionaries. This is due to the fact that most of the actors portraying Tongans were actually from New Zealand or the Cook Islands. A lot of the Tongan dialogue is also mispronounced throughout the film.
Elder Groberg has a Book of Mormon beside his bed. This particular Book of Mormon has the phrase "Another testament of Jesus Christ" on the cover. This phrase did not appear on the Book of Mormon until the mid 80's.
The film starts with a dance at Brigham Young University in 1953. A couple sing the song "Rip it Up". That song was first recorded by Little Richard in 1956.