I discovered we could get World of Hyatt points at both of these Mr & Mrs Smith hotels. They are both reasonably priced and get good reviews all around. Anyone stayed in either of these?
We never stay in hotels in Athens, so I don't really have an opinion on the hotels themselves. However, I do know their locations. From that standpoint, the Monastic is in the southwest part of the Mets neighborhood on Karea Street, which gets quite a bit of traffic (we sometimes caught the bus at the stop right in front of the hotel when we stayed in that area). Aside from that street, it can be a nice area, not too touristy, but a bit far from the main sights and with limited public transportation options.
I assume, though, that the rooms face the opposite side of Karea Street, perhaps with a view of the Acropolis.
At the Emerald, you're right in the heart of touristy Athens. From there, you can easily see and do almost everything with just a little bit of walking, and you'll find plenty of bars, shops, and restaurants. We've stayed in that area before too.
So, looking strictly at location, it all depends on the vibe you're looking for and how much walking you want to do.