The main difference between the supine and the future participle, as I see it, is that the supine is unambiguous about its expression of purpose, whereas the future participle allows for a wide range of meanings, of which purpose/intention is one. Allen & Greenough (§499) give this summary of the range of meanings (with examples):
- Likelihood or certainty (so translated as, e.g., 'about to,' 'likely to,' and even 'destined to,')
- Purpose, intention, or readiness ('in order to,' 'intending to,' 'ready to')
- Apodosis ('then' clause of a condition)
So, the idea of purpose may have to be teased out from, e.g., the idea of likelihood – or both ideas may conceivably be active at the same time, I suppose.
Moreover, as Allen & Greenough note (§498), the use of the future participle for anything except the future periphrastic (with a form of sum) is rarely found except in 'poets and later writers.' So genre and period are also factors in the choice between supine and future participle.
A particular author's personal style would also be a factor. Although A&G use Pliny as an example only for the last usage (dedit mihi quantum maximum potuit, daturus amplius si potuisset [Ep. 3.21.6], 'he gave me as much as e could, ready to give me more if he had been able,' where daturus is equivalent to dedisset), they could easily have cited passages from his letters for the others too. It was from reading Pliny that I gained a true appreciation for the expressive potential of the future participle, and I associate them strongly with his prose style.
And obviously, the supine can't be used unless there's a verb of motion.