While the technicians add a block to the "perfect prison", it seals itself so that it appears seamless. In the next frame the seam is clearly visible.
When Nixon is telling the NASA security men to let the Doctor go he tells them, "I am your Commander-in-Chief." The President is the Commander-in-Chief of the United States Armed Forces, but not any civilian organizations. NASA does some work for the military but is completely civilian and separate from the military.
During the moon walk broadcast, the voice-over states that the whole world watched Apollo 11 lift off five days earlier. Apollo 11 lifted off on Wednesday, July 16th, and the moon walk took place on Sunday, July 20th (four days later, not five).
When the Doctor is confronting the Silence at the moment Neil Armstrong sets foot on the moon, one can hear the words "You got a bunch of people about to turn blue..." spoken by one of the NASA ground controllers (CAPCOM). This was actually said several hours earlier, when the lunar lander touched down on the moon.
The implant the Doctor injects in each persons palm is of sufficient size to cause a not only a reasonable degree of motion restriction but also severe pain when trying to move any part of the hand - even something as small as a staple can cause agony due to nerve/tendon friction.
Nixon and Canton's discussion about marriage suggests that Canton marrying someone black would have presented legal problems. Although laws prohibiting interracial marriage throughout the United States were invalidated in 1967 (two years prior to the events in the show) by the US Supreme Court in the case 'Loving v. Virginia', the actual issue in this case (as explicitly stated by Canton) is that he wants to marry a man, which would not be legal until decades later.
The "live" TV broadcast of the Apollo 11 liftoff and landing have shots that were not available at the time they occurred as they were from cameras on board the vehicle.
In the opening scene Amy Pond is chased and cornered by two Jeeps.
The black one, a JEEP CJ-7 was not made until 1978 and the model shown wasn't available until 1984.
When the person in the Apollo space suit slides open the solar visor of the helmet, there is a hissing sound effect of air escaping. No such sound can occur since the solar visor is not pressurized. The helmet with the sun visor was just extra protection for space and moon walks. Only the "fishbowl" helmet worn beneath it was pressurized.