Let me say to start that I'm no JavaScript expert. I don't write JavaScript full-time (we're a small company; I don't do anything "full"-time), though I do like the language and what it can do for web-based client-side programming.
I've written non-trivial apps for the same backend, first in Ember, then in Angular. The Ember app is the consumer-driven frontend, while Angular runs the administrative interface. It wasn't my intention to use two different frameworks, but we started the Ember one when Angular was really just starting to gain traction. A few months later on the advice of a friend, I thought I'd give it a shot.
The biggest problem between the two isn't really anything about the frameworks. They're very different beasts, and they both have their strengths and weaknesses. The noticable difference out the gate is the documentation. Ember is a mess. It's older, and things keep changing, so documentation gets out of date. Also most of the tutorials and cookbooks online are all for older versions, and that can be very confusing when you're starting out.
I personally find Ember's patterns to be confusing at times, but I also have this hunch that there should be a point I get it and it will all make sense. For example, there's controller, which should manage representations of your models for UIs, and there's views, which manage events in your templates. Right now that's where I always get lost. I have no idea where to put anything. When you click on a button, is it the controller or the view that handles it? The answer appears to be "yes, depending...".
I think this confusion comes from Ember's philosophy: it's trying to be a desktop-like UI kit for web development. That works 80% of the time, and the other 20% introduces "fudge factors" like this.get and this.set. There's also too much magic happening in Ember, which makes it difficult to learn. Things happen based on the names of objects and templates, and then you hook into this magic to override certain steps. Since I don't spend all my time in Ember, when I go back to look at code I've written previously, I'm often lost at where something was updated or triggered.
Angular on the other hand is very straightforward as to where things go. For example, people mention directives. You start off by hooking events and values into your templates in the controllers. After a while, you'll start migrating that code out of controllers into directives, so you can reuse it in many different templates. As your application grows, your templates become filled with reusable directives and your controllers get simpler and simpler. It's a very nice philosophy.
Also, Angular doesn't buck the web workflow as much as Ember. Like I said, Ember is trying to be a desktop-like UI for the web, while Angular feels more like you're just transferring your older, do-it-on-the-server templates to JavaScript. This gives you a lot more flexibility with UIs and animation, but it doesn't completely break the browser, which is why I think Angular doesn't have as many edge cases to deal with as Ember.
On the downside, Angular has this initial easy learning curve, which gets steep later. The basic stuff is easy, and you'll have a web site up in a day or so, vs. a couple weeks to figure out the basics of Ember and get something meaningful online with that. However, Angular quickly gets complicated when you want to dive into complex directives and some of the more esoteric aspects of the framework.
For support, I haven't been hugely happy with Angular. There are more people using it, and there are more blog articles and tutorials out there on it, as well as lots of plugins to add features (like websockets and local storage support). That's great. But the community itself doesn't feel as helpful. Most of my posts to the mailing list went unanswered. I haven't tried Angular's IRC, so maybe that's where I should be.
The Ember folks on IRC are hugely helpful and very friendly. There is a bit of "our way is the best way", but I think that comes from understanding Ember and everything you can do with it and knowing it's probably a more powerful framework in the end. However, the discussion forums are horrible. They're using Discourse, which was written in Ember, and it's nearly impossible to find answers to anything previously posted. Because it's in Ember it's all JavaScript, so nothing is indexed on Google, and the built-in search is awful.
I would recommend if you're starting out and you don't have super, super strong JavaScript skills, you should go with Angular. You'll feel more productive quickly. Ember will be frustrating. If you're an advanced JS developer and you've built desktop UIs, maybe Ember is more your cup fo tea.
Also take a look at http://yeoman.io. It's by far the easiest way to get an Angular "build" system going. Yeoman supports Ember too, but like everything else with Ember, it's a bit tricky (for example, updates to the templates will crash grunt server if there's an error).