Linux

Tmux allows you to reconnect to a session, and helps guarantee that you will always be able to get back to your long running processes. For important long running processes, I still use tmux with mosh, because if the mosh client is killed (or you're trying to "re-attach" from a different device, mosh won't let you "re-attach" to that "session".

Mosh allows you to roam, and suspend your machine, and whenever you resume it again, whatever network you're now on, the connection is basically instantly re-established. You can often roam from WiFi to cellular data without even noticing. (Great when working from a phone, or just a laptop)

In my opinion, they are mostly orthogonal (and complementary).

Here's the list of features from the home page. I've added my own comments after '*'. If there is no '*', then the feature doesn't exist for tmux (because it's outside the scope of tmux):

Change IP. Stay connected. Mosh automatically roams as you move between Internet connections. Use Wi-Fi on the train, Ethernet in a hotel, and LTE on a beach: you'll stay logged in. Most network programs lose their connections after roaming, including SSH and Web apps like Gmail. Mosh is different.

Makes for sweet dreams. With Mosh, you can put your laptop to sleep and wake it up later, keeping your connection intact. If your Internet connection drops, Mosh will warn you — but the connection resumes when network service comes back.

Get rid of network lag. SSH waits for the server's reply before showing you your own typing. That can make for a lousy user interface. Mosh is different: it gives an instant response to typing, deleting, and line editing. It does this adaptively and works even in full-screen programs like emacs and vim. On a bad connection, outstanding predictions are underlined so you won't be misled.

No privileged code. No daemon. * Same for tmux, but that's less interesting since tmux is not a network service You don't need to be the superuser to install or run Mosh. The client and server are executables run by an ordinary user and last only for the life of the connection.

Same login method. * Not really relevant to tmux, which doesn't handle auth Mosh doesn't listen on network ports or authenticate users. The mosh client logs in to the server via SSH, and users present the same credentials (e.g., password, public key) as before. Then Mosh runs the mosh-server remotely and connects to it over UDP.

Runs inside your terminal, but better. * This is common to both Mosh is a command-line program, like ssh. You can use it inside xterm, gnome-terminal, urxvt, Terminal.app, iTerm, emacs, screen, or tmux. But mosh was designed from scratch and supports just one character set: UTF-8. It fixes Unicode bugs in other terminals and in SSH.

Control-C works great. * Tmux can help with this too Unlike SSH, mosh's UDP-based protocol handles packet loss gracefully, and sets the frame rate based on network conditions. Mosh doesn't fill up network buffers, so Control-C always works to halt a runaway process.