They are quite close synonyms, even in your example. "Biddable" has the sense of letting others direct you, whereas submissive has a more negative connotation of offering no resistance. A biddable person probably has a more constructive attitude to authority than a submissive one. Perhaps you could say that someone who is too biddable is submissive. If someone does everything you ask because they are submissive, this is worrying. In practice, biddable is very rare and I suspect old-fashioned. A common synonym is "docile".