We present the dynamic modeling, analysis, and control design of a Planar-Vertical Take-Off and Landing (PVTOL) underactuated aerial vehicle equipped either with a rigid- or an elastic-joint arm. We prove that in both cases the system is exactly linearizable with a dynamic feedback and differentially flat for the same set of outputs (but different controllers). We compare the two cases with extensive and realistic simulations, which show that the rigid-joint case outperforms the elastic-joint case for aerial grasping tasks while the converse holds for link-velocity amplification tasks. We present preliminary experimental results using a actuated joint with variable stiffness (VSA) on a quadrotor platform.