I would guess that "greater relationship maturity" involves being committed to the person you are married to, through thick and thin, through all the trials and tribulations which are inevitably a part of marriage, and even through less sexual desire on the part of the wife.
Now....just because somebody exhibits "greater relationship maturity" doesn't mean that somebody is perfect. This person might only be "mature" within the context of his relationship. He might be a total boob otherwise.