In this paper we provide a new interpretation of the Willmore energy as the curvature of a quaternionic connection that has a clear geometric interpretation in terms of mean curvature spheres rolling over the surface. Building on this interpretation, we show that the Möbius invariant discretization of the Willmore energy introduced by Bobenko can be interpreted as the curvature of a discrete connection defined by rolling the edge circumspheres. We describe how a choice of discrete mean curvature spheres produces a Möbius invariant discrete Willmore energy. In this way, we define a new discrete Willmore energy for surfaces built out of spherical pieces. To facilitate the computations we describe a new quaternionic description of the conformal three-sphere, along with realizations of the spaces of circles, spheres, and point pairs in Euclidean three-space. It is obtained by specializing the quaternionic projective model of the conformal four-sphere.