While questioning Matthew Harrison Brady on the stand, Henry Drummond refers to Charles Darwin's book as "The Origin of the Species". The book's title is actually "On the Origin of Species" (or, more popularly, "The Origin of Species").
The film is set in 1925. During Drummond's cross-examination of Brady, he uses the word "sex", not to mean "gender", but as a shorthand for "sexual intercourse". The first known example of this usage was not until 1929.
E.K. Hornbeck (analogous to H.L. Mencken) calls the town "the buckle of the Bible Belt". Although this was one of Mencken's signature phrases, he did not start using it until 1926, one year after the events depicted.
During the trial, Hornbeck hands Cates a drawing of a stick figure swinging on a gallows. The defendant wrinkles the paper in his hands immediately. The scene changes and the defendant is holding a cup and not the piece of paper.
During the courtroom examination of Matthew Harrison Brady by Henry Drummond, Drummond shows Brady a 10 million-year-old rock, which he places on the judge's desk. Later in the scene, the rock is back in his hand, and when he dismisses Brady, he places it back on the desk without ever having picked it back up.
When Drummond is inquiring about the story of Jonah, he flips to a book mark towards the end of the Bible and opens it. The Job story is in the Old Testament, but the section Drummond flips to would be towards the end of the New Testament.
At approximately 1:07, Spencer Tracy's characters refers to latitude and longitude and states "minutes, degrees, and seconds". Though technically correct, it is unusual for people that are familiar with such to list in that order; typically, logically/mathematically when listing such, degrees, minutes, and seconds are stated as "degrees, minutes, and seconds" because that is descending/hierarchical order.
When the Bradys walk toward the hotel porch after the revival meeting, the lights and buildings of the town are clearly seen behind them. But when Drummond leaves Brady on the porch a few minutes later, the scene behind them, facing the same direction, is completely blank beyond a few bushes.