Harry Shaw, Dictionary of Problem Words and Expressions (1975) has this entry for the two words:
exhibit, exhibition. An exhibit is a display of items or a collection of articles in an exhibition. An exhibition is a large-scale display, such as a fair, an exposition, or an art showing. One or more paintings by one artist might be an exhibit in an exhibition of modern art.
Merriam-Webster's Dictionary of Synonyms (1984), which treats exhibition and exhibit as members of a group of kindred nouns that also includes show, exposition, and fair, generally agrees with Shaw:
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) {the annual exhibition of the Academy of Fine Arts} {an exhibition of Navajo blankets} {a cattle show} {an industrial exhibition} Exhibit typically denotes an object or collection displayed by a single person, group, or organization in an exhibition {our club had a fine exhibit in the school fair} 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 co-extensive 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.
The confusing thing here is that exhibit can refer to a single item or to multiple items—to the extent of being (as MW puts it) "co-extensive with an exhibition." And since exhibition can refer to "any [public] display of objects of art, manufacture, commerce, or agriculture..." there is, as Philoto's answer suggests, plenty of room for overlapping usage of the two terms.
Consider, for example, the English translation of Modest Mussorgsky's piano suite, Pictures at an Exhibition. The suite consists of ten unique movements (as well as a recurring promenade), each dedicated to a different art work by Viktor Hartmann. But the ten art works were displayed as part of a special memorial presentation of more than 400 pieces by Hartmann. A 400-piece presentation of one artist's work certainly qualifies as "a large-scale display" and "an art showing"—Shaw's core definition of exhibition; but it also serves as a good example of a "one-man show which would be at once an exhibition and an exhibit of his work," in MW's formulation. And, of course, the 400-piece memorial art show was also "one or more paintings by one artist," which Shaw characterizes as an exhibit.
One point that neither Shaw nor Merriam-Webster addresses satisfactorily in discussing the possibility of "an exhibit in an exhibition" is the implicit physical proximity of the "display of items or collection of items" constituting an exhibit in its collective sense. If the ten art works that Mussorgsky focused on in his piano suite appeared as a distinct, coherent subgroup within the larger exhibition, it would be appropriate to refer to them as "an exhibit in the exhibition." But if (as seems far more likely) they were pieces scattered across the 400-piece display that Mussorgsky happened to find especially intriguing, it would seem more proper to characterize them as individual "exhibits in the exhibition."
Thus, arguably, the English translator of the title Pictures at an Exhibition might have translated it with equal technical accuracy as Pictures at an Exhibit, Exhibits at an Exhibition, or even Exhibits at an Exhibit—although not, in this particular case, as Exhibit at an Exhibition.