Not possessing the power to vote, I merely want to express my opinion:

The original version of the question was quite localized. Now, it is generalized in one way (essentially math background for a CS).
There is however another generalization that I would actually find more interesting while it is more local. It is of course unclear to me whether the questioner does feel differently or just wanted to 'escape' the local nature.

The generalization I mean, is something along the lines (the formulation I give is of course a crude one):
what 'surprises' might one experience when taking Part III after undergraduate studies in the US (I assume the questioner did his undergraduate studies in the US)?

As documented by some of the comments on the question, including one from Yemon Choi, and some brief remarks I heard once from a person in that situation it seems to me there are some. And I mean in no way somehow 'personal things' but purely technical ones; for example I got the impression that the way grades are determined / the style of the exams is quite different. Now I am neither in the US nor the UK academic system, but good responses to this of the form "In a US context you typically would expect A. But it will be B." would be interesting to me.
In this way I will be informed about both the way A and the way B, and chance are that to *me*, still from another system, actually C would be the most natural expectation.

While I personnaly and presently have no concrete need for this information, I would still find it interesting from a general point of view, and it might actually be useful at some point, say, if an undergradute should ask me for advice on studying in the US or Cambridge or alike.
(At the moment I am not in a position that typically students ask me for advice on this, but some years down the road it seems not that unlikely.)

Moreover, I am under the vague impression that US students doing Part III are somewhat common. So, in this sense the question is not a 'unique' one.

Very generally: some questions of this type refereeing / journals / relation PhD-student advisor and so on, have produced answers that were interesting to me. As regarding preceisly these subjects I feel it is in another way not easy to get opinions from people except those one knows very well. What I mean is: while to ask somebody one hardly knows or does not know at all a technical mathematical questions, say by email or at a conference, is not a big problem, I cannot imagine to ask such a person how they proceed when refereeing a paper or when they are co-authors on their students papers.