Revealing mistake: At the end of the episode there is a side shot of the Doctor leaving the TARDIS in which you can see inside, but the control room isn't visible, just the side of the police box, with light from outside shining through the window. (00:38:00)
Continuity mistake: During the opening montage, showing the Slitheen incapacitated by electrocution, when Margaret drops Indra's body and stumbles back, a close-up of the body on the floor shows her legs without the pulsing electricity covering her entire body, which is visible in other shots.
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)
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 mic is reflected on his goggles. (00:05:25)
Character mistake: The Doctor comments that the destruction of Earth will cost 5 billion lives. At the time of the episode (2006, one year on from when Rose met the Doctor), the population of Earth was around 6.5 billion. (00:25:10)
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.