Learn English – Differences between serendipity, and fate, destiny, fortune and coincidence

From New Oxford American:

dark horse |ˈdɑrk ˈˈhɔrs|
noun
1 a person about whom little is known, esp. someone whose abilities and potential for success are concealed : [as adj.] a dark-horse candidate.
2 a competitor or candidate who has little chance of winning, or who wins against expectations : a preseason dark horse as the nation's top collegiate football team.

sleeper |ˈslipər|
noun
3 a movie, book, play, etc., that achieves sudden unexpected success after initially attracting little attention, typically one that proves popular without much promotion or expenditure.
• an antique whose true value goes unrecognized for some time.

From Etymonline:

dark horse
in politics, 1842, an image from horse racing, in which dark is used in its figurative sense of "unknown."

Moonraker is called a "dark horse"; that is neither his sire nor dam is known. ["Pierce Egan's Book of Sports," London, 1832]

sleeper
[...] Sense of "something whose importance proves to be greater than expected" first attested 1892, originally in Amer.Eng,. sports jargon, probably from earlier gambling slang (1856) sense of "unexpected winning card."

Here's some generalizations gleaned from the above:

• dark horses are usually people; sleepers are usually things.
• dark horses are intentionally kept unknown; sleepers are simply not promoted.
• a dark horse might achieve success; a sleeper already has—surprisingly.
• dark horses' limelight often decrescendoes; sleepers' crescendoes.

As to your questions, I'd say they're not quite interchangeable. Knowing their different connotations will likely prove one to be the better word choice.

A dark horse might also be called an enigma, or simply an unknown; while surprise hit, cult classic, and box-office success are other phrases often applied to sleepers.

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.