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.
Continuity mistake: When little Amy looks out the window to see what makes the noise you can see the yard is out of her window (it's a little blurry) but when you look through the window properly it isn't there.
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)
Revealing mistake: When Jex's ship explodes, we see the smoke drifting over town. But it's really obvious that the smoke is coming from the next street, not miles away where Jex's ship is. (00:42:06)
Revealing mistake: In the Torchwood building, a large group of Cybermen start shooting at the Daleks and the Genesis Ark. The Daleks shoot several of them, and you can see a Cyberman that, thanks to hilarious CGI, just collapses like a sock without any physical regard that he is wearing tough metal leg armour, is floating a little off the ground, and falls way too fast than physical laws could allow. (00:25:15)
Silence in the Library (1) - S4-E8
Visible crew/equipment: While the Doctor tries to get into the data core (after Proper Dave tried) it cuts to the telephone still ringing, and when the young girl asks her dad why he's not answering the phone, just as he tells her that the phone is not ringing the boom mic dips into view at the top of the screen. (00:19:40)
Other mistake: The Doctor says: "He's taking the suspended-animation chamber to the church." The subtitles say "[...] to the judge." (00:15:30)
Other mistake: Clara says "The dark, the sword, the forsaken, the temple." The subtitles say "The Doc [...]." (00:22:10)
Other mistake: The Fisher King says "The Time Lord... Lied." The subtitles say "The Time Lord... Night." (00:35:00)
Other mistake: The line spoken is "You mash up Vikings to make warrior juice." The subtitles say "You must have [...]." (00:12:30)
Revealing mistake: In several episodes of season 9, starting at "Under the Lake" and continuing through "Before the Flood", "The Girl Who Died" and "Sleep No More", the same white hallway model is used. For the first two episodes, this makes sense, since they play in the same location. However, the same hallway is used in "The Girl Who Died", when the men from Ashildr's village get blasted to death, and again in several scenes in "Sleep No More" when the characters run through the hallways. Also, it seems the control room from the underwater research center, where the characters hide from the ghosts in "Under the Lake" and "Before the Flood", is reused as the white room in "Hell Bent" when the Doctor pulls Clara out of her timeline, though in "Hell Bent" the room is nearly empty and the room in "Under the Lake" and "Before the Flood" is full of equipment.
Revealing mistake: At the end when the Doctor makes it snow, in the wide shot you can see the snow blows in from the left side, showing that they used a snow machine just off-camera. (00:54:30)
Revealing mistake: When the Doctor goes to toss the trio of Christmas balls up, the two times after the first toss are re-used shots of the first toss. His movements of his right arm to bring up the controller are too exact to be coincidence. (00:50:15)
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.