Some journals, such as many of the EGU Copernicus publications, publish the initial manuscript and the (anonymous) peer reviews and author replies along with the full manuscript. When reviewing for such a journal, should I review any differently than when I write a review for a classical journal where only the editor and authors should be expected to ever see the review?
In theory we should be as thoughtful and careful regardless. In practice I found that I paid more attention to correct spelling and formatting for reviews I knew were going to be available online. I think the real benefit of publishing the reviews is avoiding the senselessly mean reviews and promote a more mature academic style.
One possible factor: the audience for your referee reports will be wider, which means people with less background may read them. When I write referee reports, it is often from one expert to another -- so it would be easy to say, for instance, "As you know, Fact X is true..." without a citation. This might be a bad habit in regular refereeing, but if the reports are public, it's worth being a bit more careful on that.