Continuity mistake: When Andie goes out to get the oxygen from Daedelus she cannot reach it because her helmet won't let her through the structure of the space station, so Max offers to go and help her because he is smaller, but the only suit they can find is an adult one. After a lot of struggling they manage to make the suit smaller and he fits through the structure to get the oxygen but his helmet is the same size as Andie's so although his suit was smaller, he should not have been able to fit. Also notice how long her belt is when Max emerges from the ship. She would have to have about 30 ft of belt to wrap that much around him.
Other mistake: When they first get into space and Tish removes her helmet her left earring is free floating, which would be expected in zero gravity. But after this first shot, her earrings hang down as effected by gravity. Also all three of the girls hair hang down with gravity the whole time they are in space.
Factual error: When Control realizes that the kids are communicating with Morse code, they get someone to translate. He reads out "Request alternate landing White" in the space of 9 seconds. The lights and sounds from the CXT switch are not going nearly fast enough for him to have gotten all that.
Deliberate mistake: The filmmakers wanted a scene where Andi is startled by Max as he spacewalks behind her at Daedalus. Given that the only way to communicate in spacesuits is through radio, and the two suits would presumably been on the same channel with the shuttle, Andi would have known Max was coming. Further, even if they were on different channels, why wouldn't anyone tell Andi that Max was coming?
Plot hole: The shuttle achieves its accidental orbit, and in need of oxygen they decide to go into a higher orbit to rendezvous with the space station and borrow some tanks stored there. If the shuttle is as ill-prepared for flight as the ground controllers keep saying it is (as it lacks backup oxygen and a communications system), it is highly unlikely the shuttle would have enough fuel to pull off the maneuver to enter a higher orbit (especially given how much fuel was burned off during the test before the accidental launch was triggered). Typically shuttles were launched during specific time frames (launch windows) to enable them to achieve the necessary orbit for their mission directly from launch (such as going to the International Space Station). One of the reasons a damaged Columbia, for example, couldn't unload its astronauts at the ISS was that, aside from not having a docking module, is that it was in a different orbital plane and didn't have the fuel to speed up to the ISS's orbit (which, it is said, would have been roughly equivalent to the fuel needed for takeoff). And even if the orbiter did have enough fuel to pull off the orbit adjustment, it just raises the question of why NASA felt the need to mount the shuttle on two fully operational SRBs and give it all that extra fuel when all they wanted to do was test the orbiter engine (for which empty mockup SRBs would have sufficed if the test really needed to be done on the pad). What is the point of fully mounting a shuttle if it's not for a mission to the point where you don't bother to install life-support or communications?
Factual error: During launch the Shuttle cabin is rather noisy and rattling due to vibrations from the solid rocket boosters and the main engines. After Andie and the Space Camp kids have jettisoned the solid boosters and the external tank, the cabin is still played as noisy and shaky. However, at that point the main engines are shut down and the orbiter is in freefall: there should be no noisy vibration.