Pro Flux architecture pattern
Flux is a platform agnostic pattern which can technically be used with any application or programming language.
One of Flux' main features is that it enforces uni-directional data flow which means that views do not change the data directly.
With React this is useful because this way it's easier to understand an application as it starts getting more complicated. By having two-way data binding, lead to unpredictable changes, where changing one model's data would end up updating another model. By using the Flux architecture, this can be avoided.
Con Not a complete solution
React does not do everything for the developer, it's merely a tool for building the UI of a web app. It does not have support for routing or models, at least not out of the box. While some missing features can be added through libraries, to start using React and use it in production, you still would need to have experience, or at least a good grasp on what the best libraries to use would be.
Con Large file size
react.min.js is 145.5KB in size. It's much larger than some other libraries that offer roughly the same features and it's almost the same size as some MV* frameworks such as Angular or Ember that offer more features out of the box.
Although, it should be mentioned that sometimes having a smaller library may force developers to reinvent the wheel and write inefficient implementations on features that React already has. Ending up with a larger application that's harder to maintain and/or that has bad performance.
Con You have to learn a new syntax
Requires learning a custom syntax, JSX, that has some gotchas and introduce complexity, a steeper learning curve, and incompatibility with other tools.
Though you can opt out from JSX and use vanilla JS instead. But that is not recommended since it adds a lot of unneeded complexity which JSX tries to avoid.