It's not only cleaner, it's also an excellent architecture to make responsive applications. Less information needs to be exchanged with the server - the formatting (views and controllers) being on the client, you only need to exchange the data being manipulated on the REST interfaces.
No full page reload - the server sends a static HTML file upon the first request, then the JS client handles the interaction with the user, only remodeling the portions of the DOM that changes between pages. And, better still, Backbone.js takes care of a large part of the work required to synchronize data with the server. Sweet!
However, when I recently started to learn about Backbone, I realized it doesn't help to handle relationships between models. Most non-trivial applications need this - forum threads each have a series of comments, billing invoices have several items to charge for...
If you're reading this, you've probably found out about backbone-relational after reading a few threads. But the documentation is sparse, and it's hard to see how to use it practically. How do you structure your views to represent the relations between models? How do you update or push relational models to their corresponding REST APIs?