Greetings, they are similar, indeed, but describe somewhat different actions from somewhat different angles. One who is flattering (we may call him a flatterer) flatters by words ; being obsequious (=displaying obsequiousness) is never good; it is an excessive concern about the will of another which has always interest for its end . I would say that using flattery is only one way of being obsequious.
New words, one handy idiom, and a 2-minute quiz — delivered to your inbox to keep your streak alive.