The reason folks asked, "docker is a screen replacement?" is because one typical use case is using screen within an ssh session where you might get disconnected. screen holds onto the session so that you can ssh back in and continue from where you left off.
$ ssh hostname
$ screen
# some time later, you get disconnected...
$ ssh hostname
$ screen -x
# back to where you were before getting disconnected.
mosh + tmux are simply amazing together. It has radically improved my life, where I keep persistent connections from my laptop to various remote machines.
My only complaint with mosh is how slow it is - my top-level 266x188 tmux session absolutely crawls on it, and it's quite common to see it using more CPU than whatever heavy task is spewing out lots of tty output. Wonderful for the common case, though.
Screen is amazing for SSH work or if you just don't want to have 50 tabs open. I've been looking into tmux lately though for the later. I just found out about it last week.
Seriously, the amount of commands linux has that most people don't know about is staggering. This link[1] was on HN just the other day.
I use tmux at work to manage at times up to several dozen sessions each with several windows/panes (I make one session for every conceptual task that I take on, and sessions live until I complete the task). In my experience it is much more robust than screen (not as many dungeon collapses, if you know what I mean) and well worth switching to.
Docker runs a whole container (with embedded OS), screen justs keeps a terminal accessible after a disconnect. I don't see how using Docker fits that usecase.
I haven't explored Docker yet. Some of my team members have been looking into it. Hoping to read up and experiment with it a bit once I have some free time.