Personally, I think of MMOs in a mostly gaming context, with some niche offerings such as Second Life or VR Chat leaning more towards the beginnings of a simulation-like experience.
Whereas with the Metaverse, I imagine the transition of traditionally physically-based tasks to a virtual or augmented reality world. These tasks could be work meetings, hanging out with friends, taking classes, along with lots of other things, including playing games, which is where the confusion between MMO and Metaverse reaches its peak. Some people seem to lump Web3 into the same basket though—as I said, lots of subjectivity with this being such a new and vague term.
The term 'Metaverse', at least as it appears to me, is so vague and broad as to almost render it meaningless. As a proponent of mixed reality I've been disappointed to see how Facebook has latched onto the term to the detriment of MR's reputation. Although, one could argue this is a bit more forgivable when considering the massive efforts they've undertaken to improve the accessibility of early VR.