While methods for dynamic tuning of surface wettability to manipulate water droplets have been widely explored for many applications including digital microfluidics, those based on dynamically changeable surface morphology have remained challenging to achieve. In this work, we present a structured shape memory polymer (SMP) surface which shows dynamically tunable surface wettability through changeable surface morphology in order to manipulate water droplets. The structured SMP surface involves a SMP pillar array consisting of nanotextured small and large pillars which can change its morphology between permanent and temporary shapes upon thermomechanical loading. Specifically, the structured SMP surface dynamically creates a surface morphological gradient and changes its surface wettability during thermally induced shape recovery of the SMP pillar array. Different wetting characteristics of the structured SMP surface between permanent and temporary shapes are theoretically predicted and experimentally verified. Based on these measured wetting characteristics, the structured SMP surface is designed to demonstrate that the morphological difference between two shapes under a water droplet overcomes contact angle hysteresis, resulting in driving a water droplet, when combined with the thermal Marangoni effect.