Barrie England's answer is unobjectionable. It's "standard": if you follow the rule it describes you will never be misunderstood or excite a sense of oddity in your interlocutors; and you will enjoy the approbation of well-regarded authorities.
That said, I think it's mistaken. I believe this is a case where people who have a profound interest in usage feel there ought to be some subtle difference between these constructions, and have accordingly shaped a rule which makes sense to them and guides their own usage.
There's nothing wrong with that; but my (admittedly unsystematic) observation suggests that at least in unemphatic contexts all the future-modal constructions occur entirely at random—not only those you ask about, I'm/I am staying and I'll/I will be staying but I'm/I am going to be staying and I'll/I will stay as well.
Any of these may be used with perfect propriety to "make a prediction" or to "talk about future plans and arrangements". None conveys a greater or lesser degree of certainty or intention (those may, to be sure, be communicated by selecting an uncontracted form and stressing the auxiliary).
In answer to your question "When would one use one over the other?" I would say "Whenever you feel like it."