Great question! For what it's worth, I'm the author and maintainer of Byobu.
Byobu is a configuration layer, originally written to sit on top of GNU Screen, but now also works on top of Tmux.
I started writing Byobu back in December of 2008, as I met up with a bunch of Screen and Ubuntu Server users at the Googleplex and found that all of us maintained our own bunch of neat/fun/useful hacks in our
~/.screenrc configurations. And we had to manually move those around between the dozens or hundreds of servers we used. We started trading tips and tricks, and I began to collect those into the original GPLv3 project called "screen-profiles". About 6 months later, a whole community had developed around "screen-profiles" and the project became much more than just screen hacks -- we had configuration utilities, live status plugins, and keybindings. So we renamed the project "Byobu", which is a Japanese word for those elegant, folding "screens", and has the added benefit of being able to more successfully Google for "Byobu $FOO" than "Screen $FOO".
With Byobu now in most Linux distributions (Ubuntu, Debian, Fedora, Arch), and functional on most Macs/BSDs and other UNIXes, it give the same look-and-feel, convenient keybindings, dynamic system status information at any terminal you might need to access.
Why not contribute back to the GNU Screen project? A couple of reasons... All of what Byobu works just as well as configuration options. None of it needs to be included in the Screen source base to be functional. Some things might work better or perform nicer if Screen included them by default, but many of the changes are very "opinionated", which are usually difficult or impossible to contribute to a 25-year-old upstream project. Also, the GNU Screen project is moving very slowly, if at all. It's 25+ years old, and hasn't had an official release since August of 2008. Every distribution is carrying huge stacks of patches just to keep your /usr/bin/screen working and secure. e.g., Ubuntu and Debian are currently carrying 19K lines of code in ~48 patches.
I learned of Tmux about 2 years ago, and really fell in love with the source code, design, interface, and active community! I've had a much easier time contributing fixes to upstream Tmux and discussing topics on the mailing list. And as a Byobu user who uses it everywhere, I wanted the same look and feel to my Tmux sessions as what I had come to enjoy in 4+ years of Byobu. So I ported all of the Byobu code to work equally well with Tmux as the backend, as Screen. As of the Byobu 5.0 release, Tmux is now the default backend, with Screen still supported in a legacy mode. Byobu now leverages many of the modern features of Tmux over Screen, including vastly improved 256-color support, UTF8 characters, and horizontal/vertical window splitting.
If you're satisfied with the default settings in Screen or Tmux, or want to write your own configuration files from scratch, then by all means, Screen and Tmux as fantastic utilities that have added many years of efficiency to our lives. If you're interested in a set of configurations that really stretch and extend what Screen and Tmux does out of the box, have a look at Byobu!