The Sontaran Stratagem (1) - S4-E4
Continuity mistake: When Martha asks the Doctor to look at her so she can point out she's not carrying a gun, he's sonicing an ATMOS device, and he lowers his hand when she speaks. In the next shot facing Martha, the Doctor's hand is raised again. (00:16:10)
The Sontaran Stratagem (1) - S4-E4
Continuity mistake: When the Doctor gets embarrassed as Donna lightly mocks him for not realizing that when she said she wanted to go home, she only meant for a brief visit, he rubs his chin with his left hand. When Ross Jenkins interrupts by saying, "Ready when you are, sir," the Doctor is now rubbing his chin with his right hand. In the shot after that, it's back to his left hand. (00:20:40)
The Sontaran Stratagem (1) - S4-E4
Continuity mistake: When Ross makes his "round the bend" pun while turning the Jeep onto Rattigan Academy's drive, the road seen doesn't match up with what's shown on the GPS display. (00:26:20)
The Sontaran Stratagem (1) - S4-E4
Continuity mistake: When the clone Martha sits up in the tank, in a shot from behind her the umbilical cord is trailing down her back. In a shot facing her, it's draped over her shoulder, only to return to being down her back when she unplugs it. (00:35:20)
The Sontaran Stratagem (1) - S4-E4
Visible crew/equipment: As the Doctor tries to get Wilfred out of the car, the camera goes to follow his movement and there is a reflection of the lighting rig and camera on the car.
Continuity mistake: When the Doctor pulls out a TARDIS key to give to Donna, the exact position of the key in his hand changes between shots. (00:05:00)
Factual error: Colonel Mace says the Sontaran ship is orbiting 5,000 miles (8,050 km) above the Earth. However, in shots of and from the ship in both this episode and "The Sontaran Stratagem", the Earth appears far too small for the ship to be that close, with the ship's true distance appearing to be well over double that. (00:08:34)
Other mistake: When the Earth's nuclear launch sites are being brought online, country by country, in preparation for a strike at the Sontaran ship, Russia is puzzlingly omitted, even though North Korea of all nations is included. (00:17:20)
Visible crew/equipment: During the warehouse battle, around when the Doctor tells Donna, "Hold on, I'm coming", the film crew can be seen reflected in a door at the right of screen. (00:28:20)
Continuity mistake: When Colonel Mace confronts Commander Skorr, he raises his gun twice; once in one shot, and immediately after in the other. (00:30:30)
Continuity mistake: When the Doctor activates the teleport, Donna is onboard the Sontaran ship, holding a cell phone in her left hand. When she arrives in the basement of the ATMOS factory, she's holding the rubber mallet in her left hand, and her right hand is empty, with the phone nowhere to be seen. (00:32:40)
Other mistake: Just before the Doctor ignites the atmosphere, Sylvia's house is getting dangerously full of gas. When Sylvia and Wilf go outside after the air is cleared, the inside of their house is completely clear. Meanwhile, back at the Rattigan Academy, there is still gas lingering in the air. (00:34:43 - 00:37:15)
Continuity mistake: Near the end, when Luke teleports and switches places with the Doctor on the Sontaran ship, Luke catches the atmospheric converter's switch twice, once in the wide shot and then again in his POV. (00:40:25)
Factual error: The Doctor sets the atmosphere on fire, but only the upper part burns. The atmosphere extends from the Earth's surface (where it is densest) upwards, and so the flames would come right down to the ground, burning everyone to death. Of course that wouldn't work with the plot, so artistic licence has been applied.
Continuity mistake: When Donna is sneaking around the Sontaran ship, she's carrying Martha's old phone, which she's using to stay in touch with the Doctor, and the rubber mallet from the TARDIS. There are times when one of the two objects, the phone more often, is missing from her hands with no explanation.
Continuity mistake: When they get out of the TARDIS at the beginning, Martha and Donna change positions relative to each other between shots. (00:00:55)
Continuity mistake: During the conversation where Donna gives Jenny her name, Jenny's ponytail keeps changing position between draped over her shoulder and hanging in the back. (00:06:55)
Continuity mistake: In the scene where Cobb is briefing his troops, the troops all have their guns, but in the other scene (after you see the Hath marching) the troops are receiving guns. (00:15:45)
Continuity mistake: When Donna suggests that she distract the next guard using her "womanly wiles", she flips her hair behind her shoulder. In the next shot, when the Doctor tells her to save her wiles for later, the hair has spontaneously moved to its former position. (00:19:35)
Continuity mistake: When the Doctor starts scanning the wall of the tunnel as Jenny asks him who he is and if he's an anomaly as well, his grip on the sonic screwdriver changes between shots. (00:22:20)
Chosen answer: 1) When Stephen Moffat took over he ignored a lot of what had been developed before (there is not in-universe answer). 2) It would have killed Rose, so the Doctor absorbed the energy. His body regenerated before the energy could do a significant amount of damage that would prevent regeneration.