The same applies to "we're not" and "we aren't". "We aren't" is usually saying that [u]we[/u] aren't, but others may. "We're not" is a strong statement that we are NOT doing whatever it is.

"It's not true" sounds better than "It isn't true!" I think writing out "it's not" puts more emphasis on the "not" part.

It's not different, but it isn't exactly the same.

There isn't any difference in denotation. Is there a difference in connotation?