I got a reply to my last bug report, acknowledging that it’s a bug and saying they’d send it on to their engineers.
The first one should only evaluate to zero if it’s zero for all a,b. The second indicates that that’s not the case. This one cost me several hours, as I searched elsewhere for why my results didn’t make sense. (The offending bit was originally in the middle of a larger expression; otherwise, I would have figured it out more quickly.)
Update: The Mathematica people have acknowledged this bug and passed the word on to their developers, presumably to be fixed in the next version.