Continuity mistake: After the TV signal cuts out due to the crash of the Titanic, when Sylvia says "It's gone dead" while searching for a channel, she moves her arm with the remote forward the same as she did in the previous, wider shot. (00:18:57)
Revealing mistake: When the Doctor and Donna run back inside the TARDIS to find the console room lit up in red and the Cloister Bell ringing, what's visible outside of the doors as they run in isn't the bright Shan Shen street the TARDIS is parked on but the dark soundstage the set was built in. (00:47:49)
Plot hole: In this episode, the Doctor is said to have died during the events of "The Runaway Bride." As a result, he was not able to prevent the spaceship Titanic from crashing in London. It is later shown that southeast England was obliterated by the explosion. This falls far short of what the Doctor mentioned in "Voyage of the Damned", wherein he frantically repeated that if the Titanic crashed, the drive's explosion would wipe out all 6 billion people on the Earth (that's the entire human race at the time of the show's airing).
Continuity mistake: In this episode, Rose's teeth look different than in all her other appearances. She also has a different accent, both for no apparent reason.
Continuity mistake: The house that Donna and her family are shown living in in Series 4 is different from the house the Doctor dropped her off at at the end of her first appearance, "The Runaway Bride." You would assume they'd merely moved - except that in this episode, Donna and her family are shown living in their Series 4 house before the events of "The Runaway Bride", on the day Donna gets the job that leads to her meeting the Doctor in the first place.
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.