Good question. There is some crossover between the two words in that an exposition can sometimes be an exhibition. An exhibition is a display of works--art, items for sale--in a museum, gallery, or public space. It can also mean a demonstration of skills or sometimes also a practice game played between two sports teams. Exposition is an explanation or description of an idea, or it can be a large, public exhibition of works. We would use exposition instead of exhibition usually because of the size and publicity of an event. So, an artist has an "exhibition" of her paintings, but several automobile companies would have an "exposition" of new products and models.