I switched to tmux when I needed to launch a lot of simultaneous jobs, categorized in three sessions (starting in "active", moving to "success" or "fail" when finished). I couldn't achieve it in screen at the time. The CLI interface of tmux is not very intuitive and I think somewhat inconsistent (maybe a bit like git?), but at least I could achieve what I wanted without major compromises.

After a quick google search my impression is that running a shell command in a particular GNU screen window is still painful.