Factual error: While flying across the U.S. before the Atlantic attempt Stewart lands and exits the plane. While doing a quick inspection of the airframe he approaches the engine and puts his hand on one of the radial engines cylinders which would have still been several hundred degrees, so should have resulted in a serious burn.
Deliberate mistake: The mechanic points out the overhead installation of the indicator compass. He says "Want to see where I put the compass? It's the best place I could find for it." The installation site was planned, not improvised. In order to read the compass in a mirror the compass was specially ordered from the manufacturer with the numbers reversed so it could only be read in a mirror. This mistake was intentional in order to claim a mirror was needed from a bit character.
Factual error: Before Lindbergh takes off a woman gives him a small mirror to use to read his compass in his plane. After he takes off the woman is seen on a train going home. In the window behind her you can see the Pulaski Skyway. Lindbergh took off in 1927 but the construction of the Pulaski Skyway didn't start until 1930.
Other mistake: When Lindbergh is forced to parachute out of a stalled plane near the beginning of the movie, the plane goes into a spin. It crosses the screen twice, first below Lindbergh's parachute and then above, indicating that Lindbergh is falling faster than the plane (which is possible, since the plane is spiraling rather than falling straight down). However (as indicated by the loud noise heard off-camera), the plane somehow manages to hit the ground first.