I just tried to approve an edit with No votes so that I could fix it. All it said was "Edit #18840572 has received one or more "No" votes, you must leave an edit note before you can approve it."

Pre-NGS, you were required to add a note at the same time as approving an edit with No votes, presumably to explain why you're approving the edit despite the No votes and presumably so that the voters are given some sort of notification that their votes are being overridden.

Since adding edit notes and approving are completely separate, as long as you've added a note at any point in the past, you can quietly approve an edit and nobody will know. See http://test.musicbrainz.org/edit/17593145 for example where I added a note before the No vote, then approved the edit without a note.

Also, if you haven't yet added a note, now you just get a useless error page with no way of providing an edit note and continuing. That just seems silly when it could just say that an edit note is required, give me an edit note field and a button to submit the edit note and continue approving, like it used to.