The antagonist and the protagonist have basically the same aim - they both want their old lives back. Or as Islington put it, their 'former glories'. But Richard and Islington take entirely different approaches, and in the end only one of them succeeds.
Richard should have suspected the Angel Islington was up to something. Well, in the book at least. He previously had a nightmare about an figure falling from the heavens with burning wings. And lo and behold, the only character who should have wings doesn't. Hmm, I don't know, maybe they were burnt off?
This isn't shown in the radio version, in which Islington spreads his wings twice, once alone and once for Door and Richard. Judging by their reaction and his comment about being confined, his wings are rather luxurious.
In the book, Iliaster doesn't take the ATM card Richard offers him, even though he's homeless and it's got 15,000 pounds on it. This is because he's also a Belower and he won't be able to get the money out either.
Very subtle foreshadowing: When Richard tries to make contact with Jessica, the elevator in Stockton's building works for him, even though the train and ticket machine in the Underground had ignored him. Islington must have sent someone from Below to check that the Angelus would be accessible for Door to use, and that person found out about it by snooping in Jessica's office at Stockton's, where the exhibition was planned. Richard could use the elevator for the same reason he could emerge from a stairwell into the broom closet of his own building: he was (unwittingly) tagging along behind someone who knew the special shortcuts of London Below, and their trail (including the elevator) was still fresh enough for a recently-unpersoned novice to traverse.
Door's solving of the Black Friars' riddle becomes much more impressive when you consider that the answer is "a key", and Door (like all of her family) has probably never used a key before.
A minor one: Jessica seems to not have noticed Door before Richard points the girl to her. It seems at first that Jessica is simply self-absorbed, but later it's revealed that Upworlders don't notice Belowers unless something brings them to their attention (as for why Richard saw Door, see WMG...)
How Door took out the assassin by using her Opening powers becomes much more disturbing to think about when you consider what Portico does to the orange to teach his daughter to Open things. Door didn't just Open the guy's chest; she possibly turned him inside out, alive.