In massive multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) systems, a challenging problem is how to predict channel state information (CSI) (i.e., channel prediction) accurately in mobility scenarios. However, a practical obstacle is caused by CSI non-stationary and nonlinear dynamics in temporal domain. In this paper, we propose a spatio-temporal neural network (STNN) to achieve better performance by carefully taking into account the spatiotemporal characteristics of CSI. Specifically, STNN uses its encoder and decoder modules to capture the spatial correlation and temporal dependence of CSI. Further, the differencingattention module is designed to deal with the non-stationary and nonlinear temporal dynamics and realize adaptive feature refinement for more accurate multi-step prediction. Additionally, an advanced training scheme is adopted to reduce the discrepancy between STNN training and testing. Evaluated on a realistic channel model with enhanced mobility and spherical waves, experimental results show that STNN can effectively improve the accuracy of prediction and perform well with respect to different signal to noise ratios (SNRs). Visualization and testing for unit root illustrate STNN is able to learn CSI time-varying patterns by alleviating series non-stationarity.