What are the differences between different, divergent, disparate and distinct?
Different is the broadest and most vague expression of difference, but generally it means a difference of a quality of two or more things. "They were of different __"
Divergent suggests two things which are "moving" but apart from each other. "They had divergent philosophies on the importance of the State."
Disparate tends to suggest quite a wide gap. Often in the context of things scattered, or quite far apart. Incomparable. "They had disparate views on the necessity of cinema."
Distinct merely indicates that two qualities or objects are not the same. They may be "similar, but distinct" or they may be "quite distinct" (i.e. you'd have to be a moron to not notice the difference).
Each has a correct usage. When in doubt go for "difference". It covers most of the ground and there's no point in using a big word if you have to ask about it.