I still use screen on a few servers but have switched to tmux everywhere else. They're quite similar, but tmux has a number of small advantages that add up.
* Better support for panes and splits. Besides vertical splits (which you can get in screen if you're willing to patch), with tmux you can for instance quickly join two existing windows together with a split, break a pane out of a split and into its own window, or arrange panes into a predefined layout.
* You also have the ability to name windows, reorder them, swap them, and generally move them around with tmux.
* With tmux, activity notifications are persistent and per-window. While screen can monitor windows for activity -- I always monitor one window that runs mutt, and another that runs irssi -- screen notifies you by flashing the caption once, briefly or until a keystroke, and doesn't mark the active window until the caption gets redrawn. This doesn't work so well if you're typing away in your editor window. ;) I got really tired of missing notifications, especially chats. Not an issue anymore with tmux, which immediately marks the active window.
* Session sharing works without having to setuid the tmux binary.
* Copy mode is mostly the same, but tmux maintains a stack of paste buffers, and keeps a viewable history of everything you've copied. One thing screen has though that tmux still doesn't (afaik) is the ability to copy rectangular blocks of text, not just complete lines.
* The scripting / programmatic interface for tmux is much nicer.
