One translation of the Seneca letter you refer to begins to suggest a difference:
Moreover, the precepts which are given are of great weight in themselves, whether they be woven into the fabric of song, or condensed into prose proverbs, like the famous Wisdom of Cato, "Buy not what you need, but what you must have. That which you do not need, is dear even at a farthing."
The way I interpret this, opus esse refers to the need for something in order to accomplish something else ("to become a successful politician, you need a lot of friends"), while necesse esse refers to things you actually can't do without ("living beings need food and water").
Ramshorn's Latin Synonyms supports this interpretation:
Opus est, it is wanted, it is necessary, because a want, as requisite or indispensable for the obtaining of some end or object. . . . Necesse est, it is absolutely necessary, of unchanging necessity found in natural causes, something which cannot possibly be avoided.
According to this interpretation, the sentence from the Symbolum Apostolorum correctly uses opus est instead of necesse est: "Whoever wants to be saved needs above all to hold the Catholic faith." But holding the Catholic faith isn't something you can't do without; there are lots of people who don't do that and yet go on breathing.