It isn't / It's not

FROM SKYCATCHER

I sent a text message using the phrase "it's not" and later I got thinking about what I'd written and went into this dialogue with myself about these two phrases "it isn't" and "it's not." Which is it?

I determined that, depending on the context, one would be preferred over the other as in:

"It's not fair," she screamed, slamming the door.
"It isn't fair," she screamed, slamming the door.

Clearly, the first example is how it would be said in true dialogue. 

Is it true? No, it's not
Is it true? No, it isn't

I prefer "it isn't" in that example. I think in this case the reason why is because of the word "is." IS it true. No it IS not/isn't

I guess this applies to other contractions as well, such as we're not/we aren't. I would love to hear your thoughts on this.

RESPONSE

I suppose you could make a technical case for them meaning the same thing, but to a good writer's ear, they don't. Here's how I personally differentiate between the two. When I want to emphasize the antecedent of the pronoun "it", I use "it isn't". When I want to emphasize the negative, I use "it's not".

So...using this sentence as a prompt:

"Look at those three bikes. I think that one's Bill's," said Tim, pointing to the middle bike.

If I want to emphasize more the bike at which Tim points, I'd write this for a response:

"It isn't," said George.

If I want to emphasize more Tim's error, I'd write this for a response:

"It's not," said George.

Pretty niggling, I suppose, but that's just me.

For your specific examples, if you want to emphasize the specific situation that is unfair or untrue, use "It isn't fair." or "it isn't true." If you want to emphasize the unfairness or untruthfulness itself, use "It's not fair." or "It's not true."