Though word-embedding is primarily a language modeling tool, it also acts as a feature extraction method because it helps transform raw data (characters in text documents) to a meaningful alignment of word vectors in the embedding space that the model can work with more effectively (than other traditional methods such as TF-IDF, Bag of Words, etc, on a large corpus). Word Embedding techniques help extract information from the pattern and occurence of words and goes further than other traditional token representation methods to decode/identify the meaning/context of the words, thereby providing more relevant and important features to the model to tackle the underlying problem.
However, from another standpoint, word-embedding models were not developed aiming to solve a particular feature extraction problem, but rather, to generalize and model the language used in a corpus to gain a semantic understanding of the words and the relationships between them. Such that, all the various corpus-specific tasks can then employ the same "library" of information which was collectively & exhaustively learnt by the embedding model. Meaning, the word embedding model learns a language model that is task-agnostic for all tasks on that corpus unlike feature extraction methods which are specifically task-oriented.
Hence, the similarity is - word-embeddings can effectively aid in feature extraction; the dissimilarity is - they're not primarily meant to extract features more than they are for modeling a language which might be an "overkill" for a particular feature extraction task on a dataset.