Because the theory of sheaves is a functorial theory, it has been adopted in algebraic geometry (both using the functor of points approach and the locally ringed space approach) as the "main theory" used to describe geometric data. All sheaf data in the LRS approach can be described by bundles using the éspace étalé construction. It's interesting to notice that the sheafification of a presheaf is the sheaf of sections of the associated éspace étalé.

However, in differential geometry, bundles are for some reason preferred. Is there any reason why this is true? Are there some bundle constructions which don't have a realization as a sheaf? Are there advantages to the bundle approach?