You can still achieve component level architecture with MV*, I guess the question should be MV* vs Flux which the answer is Flux architecture.

Breaking your page to small, reusable UI Components is the new standard now, it simplifies UI Development and will help you application to scale easier.

Disclaimer: This mostly relates to React and Redux

As Brandon said the two aren't mutually exclusive, and in fact they aren't really comparable directly. MV* describes an entire application, from the data to the presentation and business logic. Components simply describe a methodology for organizing your view layer, and doesn't necessarily address the state and logic of your application. You could do the other two aspects within your components, as they do in the React documentation, but I wouldn't advise mixing concerns this way.

If you're using a component based architecture for your view layer, you could conceivably use a straightforward MVC implementation using Backbone or something replacing the Backbone views with a React component tree. It's my opinion though that the traditional model pattern doesn't lend itself well to a component system with virtual dom rendering. Models are designed to contain and observe specific aspects of state, where as with a component system (with a virtual dom), you're not focused on what specifically changed within the state and only care about what the new state is so you can re-render your components.

With that in mind, managing your data using a Flux implementation is a better fit, because you gain simplicity by using singletons (or a singleton in the case of Redux) to store state, and forget about creating models to compartmentalize specific aspects of state. Then your business logic could go anywhere, but I would recommend using Redux (or my library ReduRx) and encapsulating your business logic in action creator functions. I wrote an article recently that talks about how to view React / Redux applications specifically outside of the traditional MVC pattern, so if you're interested in more of my thoughts check that out.

Why not both? Components and MVC aren't mutually exclusive. Usually one does MVC with components.

Component oriented architecture tends towards cleaner organization within the project and the entire application can be easily decomposed into small components. This will yield lot of files/dirs which is not such a big deal with a proper build.

MV* is talking about relationships among Model View and mostly Controller. Component architecture is talking about relationships among UI components.