Other mistake: When the Doctor tells the Atraxi "Oh, you're not the first lot to have come here, there have been so many," one of the monsters shown from their scan of Earth is a group of Ood. Even if in one of their uprisings they attacked the Earth, it would have been in the far future. The Doctor has had no encounters with them on Earth, and he knows them to be peaceful, so why would they be amongst the aliens who have threatened the planet? The only Ood who has been to Earth at this point is Ood Sigma, and he didn't come to attack. In addition, a group of Hath is also shown. These aliens have never been seen on Earth at all.
Other mistake: When Miss Hartigan is testing the earpod control, the men start off facing her. She instructs them to turn to the right. They turn 90 degrees clockwise. She then tells them to turn to the left. They turn 180 degrees anticlockwise. However, they should have only turned 90 degrees anticlockwise, and end up facing her again. What they actually do is turn on the spot, as opposed to turning left. It takes a further instruction from Miss Hartigan for them to face her again.
Other mistake: The security guards shoot away the lock on the door, which they then kick down with no real effort. But the door would still be held in place by the hinges - they don't kick it with enough force to rip the hinges out, and even if they had, we can see that there's no damage on the hinge side of the door anyway to demonstrate their forcible removal.
Other mistake: At the end, the Doctor and Rose treat Charles Dickens by making the TARDIS dematerialise in front of him. The engines begin to grind, but the time rotor (the thing in the console's central column that goes up and down when the TARDIS is flying) does not move.
Other mistake: After the Doctor tells Bill that Heather's face isn't symmetrical, when he and Bill look into the "puddle" together the Doctor explains that instead of seeing her normal backwards reflection, she's seeing her face "the right way round." The problem occurs in the next shot of the Doctor and Bill seen in the puddle, because the "WOW!" patch on Bill's jacket is properly where it should be, but note everything else is actually backwards. Things such as Bill's color-block top and her denim jacket buttons are on the opposite way. Also note the Doctor's ring has switched hands. Presumably this shot was edited, and the "WOW!" patch was blurred on the left of Bill's jacket then digitally added to the right side, to give the appearance that it's "the right way round" as the Doctor said.
Other mistake: Martha falls into the bog and Hath Peck rushes down to help her. She calls out "Help me, Peck!" but the subtitles read "Help me peg!" This is just one of many errors in this series' subtitles, which seem to have been added by listening to the dialogue without reference to the script.
Other mistake: When the Doctor finds the crystal and blows on it, it glows brightly. While it lights up his face as it should, the backs of his hands light up, apparently from the crystal. This is not possible, as the light comes from in front when the crystal is inside the Doctor's hands.
Other mistake: When the Doctor and Donna get onto the roof, after he sonics the door shut behind them, as they run for the crane for the window washing cart he puts his sonic screwdriver in his mouth, holding it between his teeth, and pulls something out of his pocket. It looks like he might be carrying Miss Foster's sonic pen - except that at this point in the episode, it hasn't yet been revealed.
Other mistake: After Jack uses Davitch's information to complete the Controller's coordinates, the Doctor uses a remote to project a holographic, transparent screen of the location in question. The image shows Orion's Horsehead and Flame Nebulae. In a shot immediately afterwards, looking through the holo-screen back at the Doctor and company as he explains what's going on, the image of the nebulae is not mirror-reversed as it should be, but is the same orientation as in the previous shot. The kicker is that the image of the nebulae has been altered, as well: it's already been mirror-reversed - in the actual sky, the Flame is left of the Horsehead.
Other mistake: When Mr. Thursday goes to see Vastra, to hire her to investigate his brother's death, he doesn't see her face (she's a Silurian "lizard woman from the dawn of time") until she lifts her veil to take a closer look at a photograph he brought her, upon which he faints. However, Vastra's veil is quite see-through, enough that her non-human appearance is obvious to the audience even before she lifts it. The same thing happens when she goes to visit the coroner later - although he doesn't faint.
Other mistake: Billie Piper is listed as playing "Rose" in the closing credits. Rose was certainly mentioned, yes, but Piper in this episode was actually sentient superweapon the Moment, who merely took Rose's form, and was in all other respects nothing like the former companion.
Other mistake: Shortly after everyone arrives on the Vinvocci ship, when the Vinvocci and the Doctor run through the door on their way to the flight deck, the Doctor comes back to get Wilf, who didn't initially follow because he was mesmerized by the sight of the Earth from orbit. It's obvious that David Tennant went just behind the door and paused briefly before coming back into the room, which is not what the Doctor is supposed to be doing.