1) Supposedly helpful if you end up at a US embassy overseas trying to get your passport replaced. Perhaps in the digital era that's less true. I'd like to think the embassy would have easy digital access to my passport, and absent that, that I would be permitted to access my email, where I keep a copy.

2) If you were stopped somewhere on the street and someone demanded your passport, a copy may be better than nothing, and losing a passport copy isn't the end of the world, although it may have its risks. You could always say (as a friend did last summer) that your real passport was stolen, your embassy is replacing it, ask them to accompany you to the embassy to verify. If they're opportunistic thugs, they're not going to go to the embassy with you.

I personally don't think there's one clear cut answer -it really depends on where you are, your comfort levels, etc. I'm sure there are places where you really better have your documents on you at all times (including parts of the US, especially for non-US citizens). If I'm in a major European or South American city, for example, I may feel safer leaving my passport in my lodging.