Screen has another potentially indispensable feature that is too seldom documented: multiuser mode. When appropriately configured, you can start up screen sessions that can be attached to by other users. Of course, there are access controls that allow you to configure who can attach and what rights they have.
In my work, this is heavily used to run processes that are intended to run perpetually, and that also have an interactive aspect. Anyone in my work group needs to be able to monitor these processes, and there are often numerous such processes running concurrently. We even have a GUI utility written in Perl/Tk that allows users to easily find, select, and open existing sessions or start new screen sessions.
I can envision other purposes, such as making a terminal session visible to all members of a 'classroom' (where the class could consist of remotely logged-in students).
Your example of 'backgrounding' long-running processes is a good one. Sometimes I will start up such a process, like a kernel build, at work, and then go home for the day. At home, I can check in by logging in with SSH, and re-attach to the screen session to monitor its progress.