Collision-resilient quadrotors have gained significant attention given their potential for operating in cluttered environments and leveraging impacts to perform agile maneuvers. However, existing designs are typically single-mode: either safeguarded by propeller guards that prevent deformation or deformable but lacking rigidity, which is crucial for stable flight in open environments. This paper introduces DART, a Dual-stiffness Aerial RoboT, that adapts its post-collision response by either engaging a locking mechanism for a rigid mode or disengaging it for a flexible mode, respectively. Comprehensive characterization tests highlight the significant difference in post collision responses between its rigid and flexible modes, with the rigid mode offering seven times higher stiffness compared to the flexible mode. To understand and harness the collision dynamics, we propose a novel collision response prediction model based on the linear complementarity system theory. We demonstrate the accuracy of predicting collision forces for both the rigid and flexible modes of DART. Experimental results confirm the accuracy of the model and underscore its potential to advance collision-inclusive trajectory planning in aerial robotics.