A twisted vector bundle is a weaker notion of an ordinary vector bundle whose cocycle condition is off by a U(1)-valued Cech 2-cocycle (the cycle data of a U(1)-gerbe) called a topological twist. We will introduce a geometric model of a differential extension of twisted complex K-theory using twisted vector bundles with connection as cycles and U(1)-gerbes with connection as differential twists. Here a U(1)-gerbe with connection is a total degree 2 cocycle in the Cech-de Rham double complex. We will give an introduction to the Chern-Weil theory of twisted vector bundles, define a twisted differential K-theory, and introduce a hexagon diagram of twisted differential K-theory.