When initially inserting in an open MH-47G, the team can be heard singing "The Ballad of the Green Berets". In fact, the inside of an aircraft like this is much too loud to hear people talking to you unless you are very close to them and practically screaming.
When Capt. Nelson is spotting for Air Support by heavy bombers, he states he needs to get closer to get the actual coordinates. In fact, if you can see the target and you have a map--which Nelson did--you can plot the coordinates easily on the map and give them to the aircraft or the Joint Air Controller. There is no need to 'get closer'.
When Lt. Col. Bowers, played by Rob Riggle, sips from a paper cup of coffee, you can tell that the coffee either wasn't in the cup at all or it was not at the level he pretended to sip out of. Because of the camera angle, we see that while he barely tips it to drink, below the rim the white inside of the coffee cup goes quite a ways down.
The opening scene has "September 11th, 2001, Fort Campbell, Kentucky, 7:45AM" when the child turns the TV to the news channel that reports "two planes hit the World Trade Center". Fort Campbell is in the Central Time Zone, which would make it 8:45AM New York time. The first plane hit the World Trade Center at 8:45AM, and the second plane hit at 9:03AM, both local time. If there was anything on the news at the time, it would have mentioned only the first plane, not both.
When the SF members are dropping in at night, they all put on PVS14 night vision devices, and you can see a green glow from the front all the way to the eyepiece. This is incorrect because the PVS14 does glow green from the front lens. The idea of NVGs are they take light from the front lens, amplify it and it gives a green image through the eyepiece. You might be able to see the green glow on the operator's eye, depending on the angle you look at him, but you will never be able to see a green glow if you look at NODs from the front.
When the teams are calling in heavy bombers to engage targets, they state the target is "danger close". Danger close is within 600 meters, and the targets in this scene appear to be about 75-100 meters away. The bombs are dropped and the enemy is taken out. In fact, the bombs would also have killed the friendly forces, as the concussion of 250-lb. bombs (the smallest carried by heavy bombers) would kill within a 100-meter radius.
The goof item below may give away important plot points.
Incorrectly regarded as goofs
It is explicitly stated that help for U.S. forces will take nine hours to arrive, but when Hal Spencer is severely wounded, a medevac helicopter arrives in minutes. This is because the radio operator radioed that they started a big engagement soon and that medevac should be on standby. The time passing between that radio call and the engagement is not shown in the movie, it might have been a long time.