Nothing, necessarily. Primary initial motivation for tmux seemed to be to create a BSD-licensed "screen" for the OpenBSD userland. In any event, tmux quickly appeared in NetBSD's package collection and I was an early user. Comparing the source code is where I saw a difference. Maybe I am just dumb but I found the tmux approach was easier to comprehend and I could edit tmux source easily whereas I found screen was more difficult learn and I never felt comfortable editing the source code. I also thought when it was released tmux's documentation was more elegant than screen's. I do not follow screen anymore. Things may have changed after screen finally got some competition.
This is not to say there is nothing one can do with tmux that one cannot also do with screen, or vice versa. There could be. I only opine that tmux is elegant, and more elegant than screen.
