tmux has a mouse switch--does this do more than that?

I think that just enables xterm mouse support, and allows for various mouse key bindings in tmux. I think the deal is that it's turned on by default, and lots of the UI elements by default are configured with mouse bindings. Not just selecting and resizing panes, you can right click on a pane or a workspace and one of those display-popups pops up with a "rename" dialog box.

I think this is why it's described as "mouse first" terminal.

Tmux mouse support also has right-click context menus, clicking between tabs and panes, scrolling in panes, selecting text etc.