Exhibition, show, exhibit, exposition, fair are comparable when meaning a public display of objects of interest.
Exhibition and, less often in strictly formal use except in art circles, show are applicable to any such display of objects of art, manufacture, commerce, or agriculture or to a display (as by pupils, members, or associates) of prowess or skill (as in gymnastics, oratory, or music).
Exhibit typically denotes an object or collection displayed by a single person, group, or organization in an exhibition, but in some uses it is not clearly distinct from exhibition or show, since the scope of an exhibit may vary from a single object to a collection coextensive with an exhibition; thus, an artist might present a one- man show which would be at once an exhibition and an exhibit of his work.
Exposition is the usual term for a very large exhibition, especially one involving the participation of many states and countries.
Fair may be equivalent to exposition or it may apply to a small exhibition of wares, produce, or stock sometimes for the promotion of sales, sometimes in competition for prizes for excellence; it suggests a variety of kinds of display and entertainment and usually an outdoor setting.