I've used tmux to manage multiple open terminals for years now. There are two reasons that really make the experience for me. Both are related to how you can detach from a running tmux session and then reconnect to it.
Restore a dropped session
When you're moving between your terminal and other applications all day, you're likely to accidentally close a window at some point. In a standard shell, that means you've just lost whatever state you had: running processes, ad-hoc aliases, vim buffers, etc. If you accidentally close a terminal window that's running tmux, no sweat. You can just open a new window, attach to the existing tmux session, and keep going!
tmux attach -t <session-name-or-index>
Pop in across machines
Since I work remotely, I have a work and personal machine on my home network. I keep them as separate as possible, but sometimes a work idea will strike me when I'm on my personal machine, or vice versa. I can easily SSH from one machine to the other, and then attach to a running tmux session. That session will have the state of my current project all ready to go. I can easily jot down a few notes or lines of code, and then disconnect. No need to worry about, say, a file already being open in another process: I can use the same vim process that's likely still open on my other machine.
Of course, tmux has so much more to offer, but I haven't seen these capabilities elsewhere.