tmux calls the individual shell instances windows. They are displayed like tabs in the status line. These windows can be shared between different sessions, so that any given shell instance can be in any number of tmux sessions used for different purposes or by different users. This allows configurations like the following example: User A: wAB, wA1, wA2; User B: wB1, wAB, wB2
As long as you don't close your session, you may even lose your SSH connection, it'll keep your state just as it was. So you can resume where you left off (via tmux attach).
As a tiling window manager, it'll make use of all the space. As you have multiple workspaces and you can resize, etc. you can adjust to see what matters most.
All of byobu's functionality is conveniently mapped to F1 to F12. It has a help menu to see keybindings and offers window tabs in an easy to interpret format. All this makes it easy to get started (can get in the way of power users, though).
The byobu abstraction layers don't pass the parameters on to tmux or screen that indicate that they should run as a login shell. This means that you can't run 'ssh -t hostname byobu'. You need to use 'ssh -t hostname bash -l -c byobu'. A second implication is that the inner shell won't know to read the .profile file instead of the .$SHELLrc file. I know of no workaround for this.
byobu adds a lot of functionality to the default tmux display. Most of that can't be implemented using the internal variables tmux provides, but requires executing external scripts.
This must be done on every update of the status bar, which happens once a second. That means that the system is performing a lot of forks and interpreting a lot of scripts for this "thin shell wrapper".
Byobu still uses GNU Screen or tmux as the backend, so from a usability perspective it doesn't add much in terms of new functionalities, instead it only adds a layer of abstraction on top of them.
Contains less code. Does not have to be backwards compatible. Two separate tools, each doing its job: session management may be used separately from tiling window/pane management.
There are the 13 shortcuts that let you accomplish the vast majority of what you'll ever need (Mod is ctrl-g by default):
Window manipulation:
Mod-c to create a window, Mod-l/h to adjust window size, Mod-j/k to change window, Mod-n to select nth window in view, Mod-enter to swap with master window, Mod-space to change layout.
Tagging:
Mod-t n to tag a window with the nth tag (1..5), Mod-T n to add/remove tag n to/from the window, Mod-v to view all windows with tag n, Mod-V to add/remove all windows with tag n to/from view, Mod-0 to view all windows.
This isn't too unexpected since it's not as popular, but it can be annoying; even Ubuntu's repo version is 3 years out of date and lacks the tag features.
mtm emulates a classic ANSI text terminal. That means it should work out of the box on essentially all terminfo/termcap-based systems (even pretty old ones), without needing to install a new termcap entry.