They are two very different technologies. And each will stay as long as the web 
exists. 

In short terms:

React offers a virtual dom to your application using a one way data binding 
flow which makes it easier to reason about the data flow in your application. 
Polymer does not do this.

Polymer is based on the same reasoning that existed before react was invented 
only this time it aims for easier component reuse for developers. Polymer is 
sponsored by Google the same company that is sponsoring web components 
specification of which you can get the most objective information about from 
https://developer.mozilla.org/en-US/docs/Web/Web_Components
Over there is states that the specification is divided into 4 parts:
1. Custom Elements 
2. HTML Elements
3. Shadow DOM
4. HTML Imports 

If you like material design, the best way to get it into your application is to 
use polymer and it's elements. 

At some point in the future you may be the one to take both to the alter and 
wed them so that react can speak web components and leverage the work polymer 
has already done in this area. 

Just my two cents. 
John.

Follow Polymer on Google+: plus.google.com/107187849809354688692
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