Pro Synthetic Events
DOM differences and spec inconsistencies are abstracted away. Bubbling works correctly. onChange fires onChange since that's what you'd expect (vs reality where some browsers that don't fire the change handler onBlur for some inputs). You also get top level events for free. You can declare your intention and React will figure out the optimal way to attach event handlers.
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 You have to learn a new syntax
Requires to learn 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.
Flagged Pros + Cons
Out of Date Pros + Cons
Con Patents license
The problem with React is its patent rider. React.js comes with a BSD license, but has a patent rider that gives you a license to React's patents. This sounds like a good thing, right? But this rider has a "strong retaliation clause" which says that if you make any sort of patent claim against Facebook, this patent license automatically terminates. This means Facebook can now sue you for patent infringement for using React. While you may think this is no worse than not having a patent rider at all, but that's not the case. If there is no patent rider, then there is an implicit grant which cannot be revoked.
If you work for a software company and your company has patents then keep in mind that by using React you are giving Facebook a free license to your entire patent portfolio.
More info on weak vs strong retaliation clauses.