We demonstrate a high-spatial-resolution fast Brillouin optical time-domain analysis (BOTDA) for distributed dynamic strain measurement based on differential double-pulse and second-order sideband of modulation. The frequency-agility probe wave is obtained from the second-order sideband of the modulated light by using the microwave signal from a wideband arbitrary waveform generator (AWG), which reduces the bandwidth requirement of the AWG by half to ~ 5.5 GHz. The differential double-pulse scheme is proposed to improve the spatial resolution while keeping the capability of dynamic measurement. In experiment, a spatial resolution of 20 cm is achieved by using a 52/50 ns differential double-pulse, and the distributed vibration measurement is demonstrated over a 50-m Panda polarization-maintaining fiber observing the vibration frequency of up to 50 Hz. With only five averages, the standard deviation of the strain accuracy is measured to be 14 με.