Continuity mistake: When Lilith runs her hand along the side of the Doctor's head to distract him so she can get some of his hair, in the wide shots her fingers are extended past his ear. In a shot facing him, however, her fingers are curled and in front of his ear. (00:34:40)
Continuity mistake: When the Doctor is persuading Shakespeare to come up with some words to stop the Carrionites with, in wide shots he has his right arm around Shakespeare's shoulder and is holding his left arm. In close-ups, they are not in physical contact. (00:38:35)
Factual error: The Globe has (and had) 20 sides, not 14.
Character mistake: David Tennant has a strong Scottish accent, and puts on the Doctor's English one. When David and Freema Agyeman are lying on the bed in the hotel, David forgets his English accent when he says, "There's something I'm missing, Martha."
Deliberate mistake: The Globe Theatre is shown staging performances at night. In actuality, they performed during the day, when there was enough light, since this was pre-electricity. This was done both to make the climax of the episode scarier and because they shot on location in the real Globe, which still shows its plays during the day, and thus they didn't have a lot of time to shoot during the day there.
Continuity mistake: In the episode "New Earth", the Doctor said that the titular planet is in the galaxy M87. Upon returning to New Earth in this episode, he says that the planet is 50,000 light-years from where the original Earth used to be. That's a difference of over 50 million light-years, since M87 is a very real galaxy in the centre of the Virgo Cluster, that far away from the Milky Way. (00:02:55)
Revealing mistake: After arriving in New New York's undercity, the Doctor and Martha find a screen playing the news. The Doctor tinkers with it and gets a video of flying cars passing a picturesque view of the city. This is a clip from the last episode set on this planet, "New Earth" - the fact that in a close-up of the screen, the TARDIS can be seen parked on the cliffs as it was in that episode clinches it. (00:03:55)
Continuity mistake: After Milo and Cheen abduct Martha, they put a sleep patch on her before entering their car. In the close-up where Cheen applies the patch, it's stuck on securely and neatly, with the top and bottom of the square pointing directly at Martha's head and chest respectively. In a wide shot immediately afterwards, the patch is stuck on quite loosely and has gone askew. (00:06:35)
Continuity mistake: After the Doctor gets directions to the motorway, when the red-haired female pharmacist leans over her counter to try and persuade him to buy some Happy, her hands change position between shots. (00:08:00)
Continuity mistake: The registration of Milo and Cheen's car is shown on the computer screen and stated aloud as "4-6-5-◊-6." However, there is a shot of Martha and Cheen looking out the window where the car's registration sticker is visible, showing a different registration: "9-9-6-γ-1." (00:13:15)
Continuity mistake: When the Doctor, after being placed on hold while attempting to call the police, talks to Brannigan and Valerie about other people he could contact, in close-ups, Valerie has her right hand resting on the top of her seat. In a wide shot, she lowers her hand, but in the next close-up of her, it's jumped to its previous position. (00:14:20)
Plot hole: Martha tells the kidnapping couple to switch off all their engines so the Macra don't notice them. Yet during this scene, the car remains hovering above ground, meaning the engines must be engaged in some way, and the Macra should still hear them. And these things don't hover when deactivated - the car was shown to be parked on the ground at the start of the episode.
Continuity mistake: When Martha, Milo and Cheen are talking in the powered-down car, the lighting changes between wide shots and individual close-ups.
Daleks in Manhattan (1) - S3-E4
Continuity mistake: When the Doctor turns the newspaper towards Martha so she can read the headline, in the wide shot, the corner in his right hand is bent back towards him. In the close-up showing the headline, the corner is straightened with his hand in a different position. (00:04:29)
Daleks in Manhattan (1) - S3-E4
Continuity mistake: When Diagoras enters the Daleks' lab, he is shown pulling on his right glove twice. (00:18:55)
Daleks in Manhattan (1) - S3-E4
Continuity mistake: When Frank is dragged off by the pig-slaves, the manhole opening into the theatre is depicted as completely open from below. When the shot switches above to the stockroom when Solomon pushes the Doctor away from the manhole, the lid is resting half over it, before it being completely open when Solomon moves to pull the lid over the manhole and close it. (00:23:14)
Daleks in Manhattan (1) - S3-E4
Continuity mistake: When the Doctor is taking the back off the radio, his grip on it changes between shots. (00:25:15)
Daleks in Manhattan (1) - S3-E4
Continuity mistake: When Tallulah is reunited with Laszlo, she puts her hands on his shoulders. Then, between shots, they jump to his neck, then the collar of his jumpsuit, then his jaw, etc. (00:39:35)
Daleks in Manhattan (1) - S3-E4
Continuity mistake: Near the end, when Martha asks the Daleks what's going on and one of them explains that they are planning to evolve to live outside their shells, the Doctor changes position between shots while he's shown listening to this, standing behind Frank and Laszlo. (00:44:30)
Daleks in Manhattan (1) - S3-E4
Continuity mistake: When Diagoras is seen being merged with Dalek Sec, he has gloves on, but when the hybrid Sec emerges, the gloves are gone. (00:45:00)
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.