Polymer is tied into the WebComponents standardization effort (also driven by google-related people). Pieces of the DOM are managed by relatively small, contained objects that maintain their own internal state and keep the DOM in sync. It achieves component state synchronization via data binding like Angular or Ember and some sort of KVO mechanism (eventually Object.observe if they're not using that already, I haven't looked at their code in a couple months).
When you write code for React, on the other hand, you basically pretend the DOM is stateless and write your code as if you're writing server side templating code and rendering the entire page from scratch. The framework takes the data structure that the render process builds and efficiently forces the DOM to match what was rendered. At least that's the central concept. There are plenty of caveats.
If anybody has more specific questions, I'm happy to answer them.