Factual error: In the scene after Zefram's ship goes to warp, when the ship is being turned around to return home, Cochrane remarks "Is that the Earth? It's so...small." He wouldn't have been able to see it, provided the distance the ship just flew. Warp 1 is the speed of light and in a 1 minute flight, the Phoenix would have travelled over 10 million miles - at that distance the earth would just appear like a bright star.
Factual error: In the scene where Troi is drunk and Riker asks if Cochrane can accept the truth, it is night. Then the scene switches to the Enterprise orbiting Earth over north America and it is apparently daytime because the sun is shining over north America, where the movie takes place.