I run tmux in vterm in emacs rather than multiple vterm buffers because as awesome as emacs is it still crashes or just becomes unusable with enough uptime. This way I can bounce emacs without losing my terminals.
I still use Screen/TMux to run servers, or any process that I need to keep running for a long time and still maintain control over it's STDIN/STDOUT.
But now that I use Emacs "shell-mode" I hardly ever use Screen/TMux anymore. I don't even use Bash anymore, I switch my default shell to Dash, because I don't need Bash's tab-complation in Shell-mode. I use Emacs's tab completion instead.