Deliberate mistake: When the rocket (a replica of an Atlas booster) launches, it makes it all the way to orbit in one piece. In reality the Atlas was a two-stage rocket, and the first stage would've been jettisoned long before reaching orbital velocity. (Undoubtedly the filmmakers decided to fudge this, to avoid dealing with the issue that the rocket didn't launch over the ocean, and the first stage would've fallen onto some innocent bystander's property.).
Factual error: In an interview before his first launch attempt, Charles explains that his flight plan is to make exactly one orbit of Earth, returning from orbit directly to the launch point, his farm in Texas. Launching from that latitude would necessarily incline the orbital plane to the equator, and Earth would rotate beneath the spacecraft's groundtrack during the roughly hour and a half period of the orbit. The craft would then land, not at the farm, but several hundred miles to the west.
Other mistake: During the credits, Jay Leno shakes Billy Bob Thorton's hand and then says "How are you Bob? Nice to see you". He should have called him Charlie or Farmer. (01:38:40)
