We compare two families of left-invariant metrics on a surface group in the context of coarse geometry. One family comes from Riemannian metrics of negative curvature on the surface itself (the metric on the surface group is given by the restriction to an orbit of the lift of the Riemannian metric to the universal cover). The second family comes from Quasi-Fuchsian representations of the surface group. We show that the Teichmuller space is the only common part of these two families even when they are viewed from the coarse-geometric perspective.