ConceptsI've been toying with these frameworks for a while (a year maybe?), trying to implement a basic character generator web app for Fantasy Flight Games's excellent Edge of the Empire RPG. It is not that the world needs such application, it is more about me finding the framework I can easily work with. Let the fight begin!
Angular.JS on the other hand feels more close to the big frameworks like good old JSP and ASP with a markup page and a code-behind class. Don't get me wrong: you will probably end up with the same ratio of generic vs. framework specific tags in your html using both. The difference is more in the approach: React builds bottom up, while Angular is geared toward the do everything type usage: it works really well if you do most of your page in Angular. While React tries to be a very good "V" in the MVC (or MVVM), Angular is a full stack MVVM framework.
I am pretty new to most of these and probably missed some real important concepts on both sides. At the end it is more about your personal preference and project needs: if you like the "build your toolbox, then your stuff" approach, React can serve you very well. On the other hand it has less to offer than Angular. This could be a blessing if your page has some kind of backbone.js like framework deeply integrated.