What is the difference between fasttext and DANs in document classification?

The first most important difference consists in the fact that when using fasttext you are training a language model, i.e. your own embedding vecotrs, while DAN is an architecture (not a language model) that require either a random initializion of embedding layers (which are then trained along with the other layers) or to use pre-trained embeddgings like GloVe (or even fasttext vectors!).

DAN is something that has become popular in some sense (even though I never saw this paper before now). Aveaging embedding vectors of single words before feeding them to a dense layer is a common practice if you need to perform some task at a paragraph or document level.

Just to add some peculiarities of fasttext embeddings, their are trained not for single words, but for n-grams. So during the preprocesing of the corpus from which the embedding are learnt, words are splitted into several chunks of character. For example:

'matter' would become [ma, mat, att, tte, ter, er]

and a unique embedding is then learnt for each chunk like 'ma' or 'mat'. The training follow the same logic of word2vec vectors, which means that from every chunk the model tries to predict context chunks. The advantage of learning embeddings for each chunk relies on the ability of these vectors to learn specific morphological features that classic token-level embeddings usually miss.

If it might help, for a good survey on word embeddings I suggest to take a look at this.