We study the rheology of systems of thermoresponsive microgels which can transition between a repulsive glass and an attractive gel state. We find marked differences between these two colloidal solids, within the same experimental system, due to the different origins for their dynamic arrest. While the rigidity of the repulsive systems depends solely on particle volume fraction, we find that the change in linear elasticity upon introducing attractive bonds in the system scales linearly with the adhesive bond strength which can be tuned with the temperature in our experiments. And while the glasses yield reversibly and with a rate-dependent energy dissipation, bond-reorganisation in the gels is suppressed so that their rupture is irreversible and accompanied by a high, but rate-independent, dissipation. These results highlight how colloids with responsive interactions can be employed to shed new light onto solid-solid transitions.