I most usually find the job of re-review to be quick and easy. You take a look at the revised manuscript and the answer to criticism and see if you major complaints have been answered. Maybe take a look at the other reviewer comments to see if they are cluing you in to something you'd missed and should be concerned about. But basically I don't typically get into a dogfight. Either the authors have taken a good stab at it and you say "good enough, accept that sucker" or they have not and....
..well there's the rub.
When they have basically blown you off and not responded to your insightful and completely necessary comments in a significant way...what do you do?
REJECT! of course. Right? They had a chance to improve the manuscript and they didn't. So you demand rejection.
But has anything really changed from the initial review to the second one? You said the first time that it should be published, given that the authors fix a couple of nagging details. And you don't know if the failure to respond to your (insightful and completely necessary) criticism is because they just didn't want to or because if they did so, there would no longer be a publishable result. Or they simply fear that if they did what you asked (add a control, boost the number of subjects, verify in a slightly different system or wtfever it was) that there would no longer be a publishable result.
If, however, they just didn't want to do any more work (and believe you me, I've been there) and basically disagree with your criticism...what next? In essence your stance should not have changed from the first round of review, right? The data could still be publishable if they would just do. what. you. suggested!
So your stance on accept/reject shouldn't change, right? Because if it does change to REJECT, doesn't this just mean you are being a petulant jerk, stamping your feet that you've been ignored? You don't have any additional information on which to base an estimate that the results are, in fact, as ... fragile...as you suspected them to be when you suggested more experiments be conducted. Maybe they are, maybe they aren't.
Your stance should be the same as the first time.