When you want to do many small real time operations against your data. The model fits well with most web sites (the unit of work for a given request tends to be quite small). Loggin also comes to mind as does some basic geospatial work. I think for most cases it does what a RDBMS can do but with less friction (you'll still use an ORM/ActiveRecord, but it'll be a much more lightweight).
Why don't you check out mongly.com/tutorial/index and get a quick feel for it?
That makes sense, but it is certainly not 90% of what I use an RDBMS for as the author stated. Especially if most of the things I do are not real time web. I will take a look at the link above, as I am curious about it. But I am still struggling for use cases.
By the way, thank you for your comment. I up-voted it.
Why don't you check out mongly.com/tutorial/index and get a quick feel for it?