boeuf f:
But I guess I'm not clear on how this practice differs from "plain noting"...
I actually don't think the two practices are all that different. The distinction between Samatha and Vipassana is a later addition and afaik doesn't appear in the Pali suttas. Inactually think that the Maps of Insight actually talk about the territory that the last tetrad of the anapanasati sutta talks about but in greater detail. apart from that, the main difference I see, and you can judge for yourself how big or small it is, is the level of samadhi applied. Anapanasati tends to be based on a strong degree of samadhi perhaps to the level of jhana (actually, i think that the real requirement would be something like Noble Right Concentration but that's another whole post) whereas noting practice according to vissudhimagga and other sources only requires momentary concentration (khanika samadhi).
Editing to add:
Mind you, I don't think a difference in levels of samdhi is all that trivial. This can imply a significant shift in focus (narrow to very narrow in samatha vs wide to very wide in dry vipassana). I can see how it could also cause a large difference in the way the dark night is perceived both because of the different focus and because of the general claiming and otherwise pleasant effects of samadhi.
Another difference is in the instructions. While noting practice mostly tells you to keep noting, perhaps with a shift in focus (narrower, wider, etc.) anapanasati instructs you to develop (or contemplate or explore) different attitudes towards the object.
-- end edit
This does bring up an interesting question, though, if we assume that both practices describe the same territory then we can say that anapanasati step 13 is roughly equal to A&P (contemplating impermanence). What then are the parallels for the next 3 lines and what do they tell us about the yogi's job past this stage? The next steps focus on dispassionate or fading away, Cessation and relinquishment.