All bag-of-words models use some kind of word vector - for a normal logistic regression / naive bayes, say "spam" and "spammer" are first and second indexed words in the vectorizer, then their word vectors are like [1, 0, 0, ....] and [0, 1, 0, ....] (length = vocab size). For both logistic regression and fasttext you get the 'document vector' by adding up their respective word vectors before applying a final linear projection + softmax to get the class predictions [1]

The insight of fasttext is to notice that these high dimension, unit vector word embeddings aren't an ideal way to learn - since they are orthogonal, during training time, a datapoint giving signal between the "spam" word vector and the label gives you no information about "spammer", or indeed any other word. Explicitly modelling and learning word vectors helps with this as now these are two points in a vector space that can be moved relative to one another.

[1] this remains true for LR if you rescale the vectors using e.g. TFIDF or use the hashing trick (a la vowpalwabbit).