Both Amber and Backbone fall into what I call the KVO frameworks category (Angular, Javascript MVC, Knockout, Batman, others). Both use jQuery for DOM manipulation and revolve around the idea of having a canonical model and having that as the center of your app. As you change the model, it fires events that you use to update the DOM. The difference is that Backbone is philosophy.
Backbone provides a basic but complete set of tools for observing model changes combined with a set of utilities that are useful but not prescriptive to make wiring up DOM events, URL handling, etc straightforward. It works with what's in jQuery/the browser. All together, it provides the organization most js apps are sorely lacking but doesn't dramatically reduce LoC over what you could do with well factored jQuery and underscore. It's nice enough that the YUI team basically adopted it wholesale for the App framework in 3.5.
Amber is all about binding. You can not only observe properties but bind them to other properties bidirectionally so that changing one causes the change to propagate through all bound values including things like values being arrays. This extends to the Handlebars templating (which was written for Amber) where you can bidirectionally bind, for example, a boolean on your model to a checkbox and to a class on a div so that checking the checkbox toggles the class without anything in your app directly manipulating the DOM. The implementation is an attribute plus two object path strings in the template. It's not panacea but since DOM/Event/State manipulation is ~60-70% of most JS apps I've worked with, you can achieve dramatic code size reductions.
The cost is that Amber is 10x the size of Backbone, the templating language is tied to the framework, and there's more overhead in understanding the concepts, how they fit together, and how to apply them to your code. When I talk to people who are writing jQuery spaghetti, I steer them towards Backbone for the simplicity and superior docs but mention that I use Amber for my own projects.
Can anyone explain the differences between Amber and Backbone, why Amber might require less coding?