Why would you use word embeddings to find similar words?
It depends on how similarity is defined. If similarity is defined as human-defined semantics, then a synset (i.e., synonym set) is most appropriate. If similarity is defined as frequent co-occurrence, then word embeddings are most appropriate. Even within semantic similarity, there are many approaches beyond synsets.

One advantage of word embeddings over synsets is the ability to automatically find similarity with multi-word term vocabulary. For example, the common word analogy - Man is to king as woman is to queen.