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exhibition is rather like a purely display and does not indicate the nature of the display, while fair is a kind of exhibition that is ratherly involves business or money
ex·hi·bi·tion
noun
1. an exhibiting, showing, or presenting to view.
2. a public display, as of the work of artists or artisans, the products of farms or factories, the skills of performers, or objects of general interest.
3. an exposition or large fair of extended duration, as a world's fair.
fair
noun
1. an exhibition, usually competitive, of farm products, livestock, etc., often combined in the U.S. with entertainment and held annually by a county or state.
2. a periodic gathering of buyers and sellers in an appointed place.
3. an exposition in which different exhibitors participate, sometimes with the purpose of buying or selling: a science fair.
4. an exhibition and sale of articles to raise money, often for some charitable purpose.