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Pro apt-get or yum install byobu
If neither tmux nor screen are already installed, installs tmux. Both screen and tmux can be installled at same time. Switch between either easily.
Pro Abstracts tmux and screen with a single user interface.
Pro Easy to get started
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).
Pro Adds OS dashboard alerts
byobu has support for OS alerts when an event happens.
Con Can't be used as login shell
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.
Con Comparatively heavy
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".
Con Adds only a relatively superficial abstraction on tmux or screen
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.