It could be said that to believe that X is true is simply to hold that X is the case. Truth is a characteristic of statements, not the world. So justification for X being true is exactly the same thing as justification for believing that X is true. It's just two slightly different ways of saying the same thing.
However, if someone asks "Why do you believe X?", what if part of the justification is based on information not accessible to the questioner? Say experience for example. In that case, I would say that the belief holder's belief may be justified to himself but they may be unable to justify it to outsiders. The question seems to be centered on the person being asked the question...why they believe.
However, if asked "What justification can you provide that X is true?" The question is implicitly asking "Why should I (or anyone) believe X is true?"
Is justification for X being true different to justification for believing that X is true?