After the Orbiter lands at the end of the mission, the scene inside the simulated cockpit has the sound of jet engines slowing down. The orbiter does not have any jet engines, and doesn't even use rocket engines during landing; it simply glides to its landing.
During the campfire scene, it shows the constellation Orion in the sky, with the crescent Moon. The Moon would never be in that constellation.
Even though the story is set in 1988-89 some of the facilities shown at the Alabama Space and Rocket Center - especially the Saturn V center - are all post 2000 constructions.
The jumpsuits used in the movie, complete with the current Space Camp patch, didn't come around until much later. During the period the movie takes place (1989), the jumpsuits were sky blue, versus the darker blue used, and the Space Camp patch said "US Space Camp" rather than the current "Space Camp".
The bus used to transport the team to the pool (the "Gambler" sing-along scene) was a newer 2000's/early 2010's Blue Bird complete with modern features, which ran contrary to the 1980's era Blue Birds that would have existed when the movie made.