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When we say,

It is not fair.

or

It is unfair.

I'm not sure enough to say whether both of the sentences have the same meaning or not though superficially, there is no difference between them but if we say,

Something is not possible.

or

Something is impossible.

then there is really an observable difference in my first language. Hence, the question - is there a difference between the preceding two sentences in English?

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The first group of sentences has also a noticeable in my first, second and third languages but I'm not sure enough to say anything about English. – Tiny Oct 29 '12 at 5:10
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An extensive corpus search might tell us that they were used in different contexts. Or it might not. – Barrie England Oct 29 '12 at 8:22

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To me the only difference is style: one word (unfair) versus two words (not fair). The same for impossible and not possible: both mean cannot be done for some reason or other.

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I'm not going say what you say is incorrect though let me tell you a joke just as an example, please don't mind. If you were told by someone, "The sun is going to rise in the west tomorrow" then what would you reply? I don't comment anything about what native English speakers would reply but we would, in all of the languages that I know (except English - not sure) reply - "It is impossible" and never - "IT is not possible" because that phenomenon is never going to happen at all - hence, the reply would be - "It is impossible". Sorry for the trouble, sir. – Tiny Oct 29 '12 at 7:57
In all of those languages, there are many such observable differences, hence - it is not reasonable is not exactly the same as it is unreasonable, it is not grammatical is not exactly the same as it ungrammatical, it is not believable is not exactly the same as it is unbelievable and so on. Again, I'm not sure about English. – Tiny Oct 29 '12 at 8:09
In response to "We would never say It's not possible": Such claims are presumptuous because they imply an omniscience that only a novelist has. Predicting what native speakers will say is based on probability & statistics. I never know what I'll say until I say it. Context decides. I might not even say it in English: I might say "Muri da yo!(Japanese for "Impossible!") to my Dutch friend who's fluent in Japanese & English. If I want to say it without implying that you're stupid, I'd say "That's not possible". If I want to be dismissive, I might say "Impossible!" – Bill Franke Oct 29 '12 at 8:56
In your next comment, you say that the two ways of denying something are not equal. I agree that this is sometimes true. Style differences can, but don't always, express speaker attitude differences. Some speakers & writers don't care about or, perhaps, understand the flux or feculence of word choice, word order, or style. Some do. What I said, of course, was just about me & my idiolect, not a claim about what other native speakers think. No trouble. You always ask good questions. I always try to give you a good, honest answer without being arrogant. I don't know it all. – Bill Franke Oct 29 '12 at 9:13
@Tiny In my mother tongue, German, I would say both It's not possible and It's impossible when someone would tell me the sun is going to rise on the wrong side. Though, I believe there's a slight difference, based on the way I would stress the words: That's impossible (, you stupid idiot) vs That is not possible (and you know that). Not sure about English anyway. – Em1 Oct 29 '12 at 9:27
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I don't think there's any difference of meaning between unfair and not fair, or impossible and not possible; but there are all sorts of other reasons why you might choose one over the other. A few examples:

  • Discourse Context — If someone says to you, “This proposition is not very probable”, you would very likely mirror his construction and say “Not probable? It's not possible!” But if he had said “This proposition is improbable” you would respond "“Improbable? It's impossible!”

  • Register & Rhetoric — If you have seen the movie The Princess Bride you may recall that Vizzini is given to saying “Inconceivable!”, and is eventually rebuked by Fezzik that “I do not think it means what you think it means.” Vizzini's use, of course, is ordinary colloquial hyperbole, and “Impossible!” is often used the same way. Accordingly, in conversation, when you mean impossible literally you may prefer to say not possible, so that you will not be understood to mean merely very unlikely.

  • Prosodic Context — The im- in impossible and un- in unfair are unstressed syllables, and don't take stress as readily as not. Small children (in my experience) always shriek “It's not fair!” rather than “It's unfair!” when someone else gets something they want—somehow the spondee is just more indignant than the iamb. In print, similarly, you might use not possible if you feel that italicizing the first syllable of impossible doesn't emphasize the negative adequately—or if, like me, you've never figured out how to emphasize just a segment of a string in these postings.

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