Nontheism was only created and continues to be used in order to avoid the negative baggage the comes with atheism due to the bigotry of so many Christians towards atheism. Thus if there is a difference between nontheism and atheism, it lies entirely in the attitude of the speak towards atheism: an atheist is a person who doesn't believe in any gods and isn't ashamed to say so; a nontheist is a person who also doesn't believe in any gods but is ashamed of it on some level and doesn't want to be associated with all those dirty atheists who won't stay in the closet.
The earliest usage of non-theism may be from George Holyoake in 1852. According to Holyoake:
Mr. [Charles] Southwell has taken an objection to the term Atheism. We are glad he has. We have disused it a long time [...]. We disuse it, because Atheist is a worn-out word. Both the ancients and the moderns have understood by it one without God, and also without morality.
Thus the term connotes more than any well-informed and earnest person accepting it ever included in it; that is, the word carries with it associations of immorality, which have been repudiated by the Atheist as seriously as by the Christian. Non-theism is a term less open to the same misunderstanding, as it implies the simple non-acceptance of the Theist's explanation of the origin and government of the world.
George Holyoake at least adopted a positive-to-neutral attitude. Today usage of non-theism is more likely to be accompanied by a hostile attitude towards atheism: people insist that nontheism and atheism cannot mean the same things and while atheism is dogmatic and fundamentalist, nontheism is open-minded and reasonable. It's the same sort of argument heard from people who are convinced that agnosticism is the only "rational" position to have.