When Buzz is shaving the Prince, there is shaving cream all over his mustache; before Buzz gets to the mustache, all the shaving cream disappears.
Inside the plane, the sun appears to be setting or rising on the horizon, but less than a minute later outside on the tarmac, the sun is much higher in the sky.
After Buzz receives the call from the White House, several newspaper stories are shown. The "Atlanta Herald" story goes for one paragraph about Buzz. The next paragraph begins, in a different type font, with a description of what was likely the newspaper's original account of an auto accident.
Although presumably unmarried, Buzz wears a wedding ring throughout the movie.