State control. The most important thing about a React app is that you can write your code to have every bit of application state inside a single variable. It gets you hot code loading, trivial testability, and session recording/replay. I can actually test the app without having to use selenium (which I hate). These aren't really benefits from React per se but from the vdom approach. My apps are in Clojurescript and don't use the majority of React's features so replacing it with an alternative vdom implementation is on my todo list for this year.
I haven't used it in anger, but Polymer is at the top of my list if I needed to write an app to integrate with a third party or something where I didn't have app state control.