What Simons-Sullivan call a structured bundle in
is a certain equivalence class of vector bundles with connection, such that the corresponding Grothendieck group is a model for differential K-theory.
The equivalence relation divided out is essentially the following:
two connections on a bundle are taken to be equivalent, if the gauge transformation that relates them induces on all possible lifting gerbes of the bundle a morphism of gerbes with connection, i.e. a morphism of the corresponding abelian differential cocycles.
Details are currently still in a subsection at differential K-theory.