The ads that Google displayed for this second search related specifically to training courses to do with AdWords, _not_ to do with the subject I was searching for.
I re-did the search for the training course and - lo and behold - no ads were displayed (as I would expect: it was in a fairly esoteric area).
Could it be that AdWords can be displayed not just based on the page in Google that you're currently viewing, but also on what you've just been looking for? Pretty clever if so.
I hadn't seen it before: dertyfern, you get the prize from me for observation!