The difference is the angle. The Facebook Close Friends is useful but is made for one to one relationships than organizations. App.net is not advising that you make an alert appear on your fans' phones everytime your band publishes anything on social media; they only want the band to use this feature when a new song is being released or they are about to perform in your area. It is publisher centric; Facebook Close Friends feature(which I love and use!) is consumer-focused. I don't think my Close Friends even know I get an alert each time they post something. In contrast I bet app.net plans to expose detailed analytics including engagement and unsubscription rates so publishers can further optimize content and reduce noise.
In that way, it sounds like a Facebook Page + Close Friends feature + letting publisher trigger notification on selective posts only.