Continuity mistake: When the Doctor sonics the payphone so Donna can call her mother, he's doing it with his right arm, then it cuts to close-up and he's using his left arm. (00:09:00)
Rise of the Cybermen (1) - S2-E8
Continuity mistake: In this episode's TARDIS scenes, the yellow padded seat is positioned so that it's on the side of the console closer to the door, so people sitting on it are facing away from the door. In all other episodes featuring this version of the console room, the seat is positioned on the side of the console opposite the door.
Continuity mistake: Partway through the episode, there's a wide shot of London, with an undamaged Big Ben visible. The inciting incident of the previous episode was a spaceship crashing into the clock tower, severely damaging it. (00:25:40)
Continuity mistake: During the Doctor and the Master's conversation in the junkyard, the Master gets dust on the legs of his jeans. Just before he lightning-jumps away, the dust vanishes. (00:35:15)
Factual error: The opening shot of the episode reuses the zoom-in from orbit originally used in "Rose." The North Pole is sunlit, as it would be at the height of summer. "The Runaway Bride" takes place on Christmas Eve, four days after the Northern Hemisphere's Winter Solstice, when the Pole would be facing away from the Sun, in darkness. In fact, this was the second Christmas special in a row to make this mistake, after "The Christmas Invasion." (00:00:05)
The Christmas Invasion - S2-E2
Continuity mistake: When the robotic Christmas tree attacks Rose, Jackie and Mickey in the Tylers' flat, it barrels through the internal window and destroys a supporting pillar separating the living room from the main hallway. Later in the episode, the pillar is seen intact. (00:13:30)
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 Unicorn and the Wasp - S4-E7
Continuity mistake: When Agatha shows the Unicorn's case of tools to everyone when they're all gathered in the parlour, the case is shown open just before she's shown opening it. (00:29:50)
Factual error: During the climax, the Moon is visible in the background, and is depicted as full. On December 24, 1851 in real life, the Moon was at first quarter phase.
Continuity mistake: At the end of the hospital roof scene, as the Atraxi are flying away, the Doctor is seen reacting to the TARDIS key in his jacket pocket heating up twice. (00:54:20)
Factual error: The Doctor says that the Earth is six billion years old, which is one and a half billion years over - it's only 4.5 billion years old. (00:21:25)
Audio problem: When Martha says "We're on the Moon. We're on the bloody Moon," her reflection is visible in the window in front of her. Her lips don't move. (00:07:35)
Continuity mistake: In the sphere room the Doctor briefs Yvonne and Rajesh on the void. As he sits down in front of them you see Jackie's feet start walking towards them but the next cut behind Yvonne and Rajesh facing the Doctor shows Jackie isn't there. (00:25:15)
Continuity mistake: When the Shade leaves Clara and she falls, her body is twisted because of the way her legs are bent with one on top of the other and her arms are out to the side. In the overhead shot of her, her positioning has changed, her body isn't twisted and her arms are more straight down her side.
Visible crew/equipment: When Amy is guesstimating Amelia's age, the camera pans to the right and into a crew member's arm. (00:13:49)
The Snowmen - S7-E7
Continuity mistake: When the Doctor is explaining what the intelligence in the giant snow globe is, though enclosed, the amount of "snow" alternates from a fair mass to a few handfuls and back several times.
Visible crew/equipment: As the black suited figure stops to look at Martha, the crew and equipment are reflected in the helmet and visor. (00:02:30)
Visible crew/equipment: After the acting Prime Minister gives orders to the sergeant and the lift doors close, the soldier turns around. As he issues orders, the boom mike is reflected on his goggles. (00:05:25)
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)
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.
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.