To give a brief, simplified answer: for a RESTful backend, at the moment, you should use Backbone.
To give a more complex answer: It really depends on what you're doing. As others have said, Ember is designed for different things, and will appeal to a different set of people. My short answer is based on your inclusion of the RESTful requirement.
At the moment, Ember-Data (which seems to be the default persistence mechanism within Ember) is far from production ready. What this means is that it has quite a few bugs and, crucially, doesn't support nested URIs (/posts/2/comments/4556 for example). If REST is your requirement, then you'll have to work around this for the time being if you choose Ember (i.e. you'll either have to hack it in, wait, implement something like Ember-Data from scratch yourself, or use not-very-RESTful URIs). Ember-Data is not strictly part of Ember, so this is entirely possible.
The main differences between the two, aside from size, are basically:
Ember tries to do as much as possible for you, so that you don't have to write as much code. It is very hierarchical and, if your app is also very hierarchical, will likely be a good fit. Because it does so much for you, it can be difficult to figure out where bugs are coming from and to reason why unexpected behaviour is happening (there is a lot of "magic"). If you have an app that fits naturally into the type of app that Ember expects you to be building though, this likely won't be a problem.
Backbone tries to do as little as possible for you so that you can reason about what is going on and build an architecture that fits your app (rather than building an app that fits the architecture of the framework you're using). It's a lot easier to get started with but, unless you're careful, you can end up with a mess very quickly. It doesn't do stuff like computed properties, auto-unbinding events, etc and leaves them up to you, so you will need to implement a lot of stuff yourself (or at least pick libraries that do that for you), although that is rather the whole point.
Update: It appears that, as of recently, Ember does now support nested URIs, so I suppose the question comes down to how much magic you like and whether Ember is a good fit, architecturally, for your app.