"I would like to see a post addressing how to handle a paper sent back by the editor for another round of reviews. Of particular interest to me: what do you do if you notice errors that escaped notice (or weren't present) in the original manuscript? What if the authors answered your issues well in the response letter, but didn't include those modifications in the manuscript? What advice would you have if the authors have clearly done the experiments and theory well, and the results are worth publishing, but the writing/figures are still not at a publishable level following their revisions?"
My answers are probably what you'd guess. I try hard to identify possible errors the first time through refereeing a paper. If I spot something on the second round that I'd missed, I try to be clear about this by writing something like "Also, on this pass through the paper, I had the realization that [blah blah] - my apologies for not catching this on the first round, but I think it's important that this issue be addressed." Again, I try hard not to miss things on the first pass, since I know how annoying it is from the author side to be hit with apparently new objections that could have been addressed in the first revisions.
I've definitely had cases where the authors wrote a great response and then made almost no changes to the manuscript. In this situation, I usually say, "The response letter was very convincing/clarifying regarding these points, and I think it is important that these issues are discussed in the manuscript itself." I would then, in the "comments to the editor" part of the report, re-emphasize this, in the hopes that the editor will push the authors about it.
If the manuscript contains good science but is written at an unpublishable level (rare, but not unheard of), I try to point this out diplomatically (e.g., "The science here is very interesting and deserving of publication, but I strongly recommend that the presentation be revamped substantially. I think swapping the order of the figures would make the story much clearer."). Again, to the editors, I might make more specific recommendations (e.g., "This manuscript absolutely needs to be closely edited by a native speaker of English" if it's full of truly poor grammar).
The basic strategy I follow is to try to evaluate the science and offer as useful and constructive feedback as possible (given that I can't spend tons of time on each refereeing assignment), in the kind of professional and courteous tone I'd like to read in reports on my own work.