I use both. Specifically, in Emacs I use ansi-term, and from there I use tmux.

There are some more configurations that I had to make, specifically for copying and pasting, but I should probably spend more time to code better integration (I would love it if Emacs could put things in tmux clipboard, so that I only use one paste command, similarly, I'd like that in tmux buffers all copying would put things in tmux clipboard instead of Emacs / terminal / tmux).

The reason for me to do it is that I have to work with my allocated resources in our datacenter, and I want to keep my workspace intact between the times I disconnect from it. Also, it's sometimes easier to work with an editor local to the filesystem where I need to edit the file vs using Tramp (esp. if the files are large). So, I typically go for vi in tmux (inside Emacs) when I need to edit something remotely. Even viewing long logs with Tramp isn't a great experience.