The exhibition can be seen as a term for an event to be displayed in the public for people to view or access. The exhibition is the noun form from the word 'exhibit,' which means to show or displace for others to see. It means 'to demonstrate,' 'to submit,' 'to show off,' 'to administer,' usually on a public display. A very synonymous word to the word exhibition is exposition.
The term 'exhibition' is usually used in art and its world. Artworks are usually placed in the museum for exhibition. Sports activities can be organized for an exhibition. On the other hand, the fair can be used as a noun, verb, adverb, and adjective with different meanings.
Fair as a noun, can mean a community gathering exhibit and probably celebrate local achievements, or an event for public trade and entertainment. An exhibition is not fair, but the exhibition is contained in most fairs.