Hablando de objetos, la diferencia entre 'exhibit' y 'display' no es tan sutil. It's a different story when the words are verbs: Ejemplos: A patient suffering from malaria exhibits symptoms typical of the disease: fever, convulsions, cold sweats. He does not display them. An arrogant mafiosi displays his wealth; he does not exhibit them. Regresando a objetos, la pantalla de un telefono celular es una 'display'. 'Exhibit', como ya dijo H, normalmente se usa en el contexto de museos y galerias de arte.

Display and exhibit can be for the same purpose. However, exhibit is used mainly in the context of a museum exhibit, and a display is also used in context of displaying artwork within a museum but the word display can be used on a more casual basis.