One example would be turning a group of statements "xyz" into "if (some condition) then xyz, else (something else)" - with a text editing interface you have to think a bit about the exact text selection you want (or enter visual mode in vi to get the lines), cut them, write the if statement, paste them in, and usually some reformatting, whereas with Tree mode (the dedicated mode for expressing meaningful edits to the code) you can do Esc-w for "wrap", type the if statement, and insert the original statements from a dedicated clipboard.

Other cases are similar, e.g. being able to just drag and drop a div into another div without thinking about the exact text selection - the benefit is having to think slightly less about the mechanics of your editor or the particular data structure (text or ASTs).

Isn't this what the Intellij/Eclispe IDEs have been doing since the very beginning? All the powerful refactoring features are based on AST manipulation.