Also (at least for me):-
Backbone learning curve: 1-2 months
AngularJS learning curve: 3-5 days
Also (at least for me):-
Backbone learning curve: 1-2 months
AngularJS learning curve: 3-5 days
Angular offers a different and more indirect set of abstractions, which means that (for me) it's harder to reason about what's going on, and it's not really feasible to look at the Angular source for clues. Angular feels like a good approach if you're willing to adopt it fully and follow its conventions, but this surely takes longer to learn fully than Backbone does.
Also, Backbone plays incredibly well with CoffeeScript, if you like that kind of thing (I do, but I understand that not everyone does).
I've been writing MVC apps in JS and ECMA script in general (eg AS 3) for a good time now, and have used other MVCish frameworks before (Dojo, Flex etc). Backbone has a solid implementation of models, views and events. If you understand the fundamentals of Javascript (knowing event listener closures are references that'll prevent GC) and write modular code you can do great things with it without any "mess".
Having not tried any JS MVC framework before, I also have the problem of thinking backbone js doesn't seem to be so good as everybody is saying. Although I have to admit that I did not spend enough time yet. But from the examples (in particular the famous Todo example), I got the impression backbone js only works fine for apps with low complexity.
But who am I to judge, having it not used yet seriously...
That being said, I see no problem at all with using them both at the same time. There is a Knockback project, which I found after I wrote a hundred lines of code helper that does exactly the same thing, and I can tell that the work great together. Backbone is good at the Model layer, it's Model ad Collection classes are very good; Knockout on the other hand is exceptionally good when it comes to Views and Controllers, almost entirely eliminating the latter.
I think that people fear the complexity of using both frameworks in one app, but in my experience it's not that bad. Everyone should give it a try.
Backbone gives you tools for modeling the client-side data of your JavaScript application, and having an interface that reacts dynamically to those changes. In Backbone, the emphasis is on the data and rich models; in Angular, the emphasis is on the HTML structure.
Fortunately, there are many good examples you can take a look at for both -- http://builtwith.angularjs.org vs. http://backbonejs.org/#examples. A longer answer to the question is available in the FAQ, here: http://backbonejs.org/#FAQ-why-backbone
There is literally a couple of days worth of reading to get up and running with either backbone or angular and the rest of the time is just standard write code -> question how good it is -> research better ways -> refactor loop that everyone should go through when picking up something new.
Ember - 10000000 years
How steep you find a learning curve is likely to be influenced by your technical history.
That said, the difference in difficulty, in my experience, is certainly not an order of magnitude.
My backbone code is very easy to understand, organized and not a mess at all. It doesn't need to be for everyone, but I think your recommendation may be misplaced.