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)
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)
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)
Chosen answer: The Master knows that deep down, he deserves death for the crimes that he's committed throughout his life, and since he regards The Doctor as his arch-foe, he expects it to be at his hands. The fact that The Doctor is still willing to forgive him for all of his crimes hurts him more deeply than death would.
Captain Defenestrator