A typical web app is mostly stateless, because of its request/response nature. The HTTP protocol is the best example of a stateless protocol. But since most web apps need state, in order to hold the state between server and client, cookies are used such that the server can send a cookie in every response back to the client. This means the next request made from the client will include this cookie and will thus be recognized by the server. This way the server can maintain a session with the stateless client, knowing mostly everything about the app's state, but stored in the server. In this scenario at no moment does the client hold state, which is not how Ember.js works.
In Ember.js things are different. Ember.js makes the programmer's job easier because it holds indeed the state for you, in the client, knowing at every moment about its state without having to make a request to the server asking for state data.
However, holding state in the client can also sometimes introduce concurrency issues that are simply not present in stateless situations. Ember.js, however, deals also with these issues for you; specifically ember-data is built with this in mind. In conclusion, Ember.js is a framework designed for stateful clients.
Ember.js does not work like a typical stateless web app where the session, the state and the corresponding cookies are handled almost completely by the server. Ember.js holds its state completely in Javascript (in the client's memory, and not in the DOM like some other frameworks) and does not need the server to manage the session. This results in Ember.js being more versatile in many situations, e.g. when your app is in offline mode.
Obviously, for security reasons, it does need some kind of token or unique key to be sent to the server everytime a request is made in order to be authenticated. This way the server can look up the send token (which was initially issued by the server) and verify if it's valid before sending a response back to the client.
In my opinion, the main reason why to use an authentication token instead of cookies as stated in Ember Auth FAQ is primarily because of the nature of the Ember.js framework and also because it fits more with the stateful web app paradigm. Therefore the cookie mechanism is not the best approach when building an Ember.js app.
I hope my answer will give more meaning to your question.