Multi-station weather prediction provides weather forecasts for specific geographical locations, playing an important role in various aspects of daily life. Existing methods consider the relationships between individual stations discretely, making it difficult to model the continuous spatiotemporal processes of atmospheric motion, which results in suboptimal prediction outcomes. This paper proposes the Continuous Diffusive Prediction Network (CDPNet) to model the real-world continuous weather change process from discrete station observation data. CDPNet consists of two core modules: the Continuous Calibrated Initialization (CCI) and the Diffusive Difference Estimation (DDE). The CCI module interpolates data between observation stations to construct a spatially continuous physical field and ensures temporal continuity by integrating directional information from a global perspective. It accurately represents the current physical state and provides a foundation for future weather prediction. Moreover, the DDE module explicitly captures the spatial diffusion process and estimates the diffusive differences between consecutive time steps, effectively modeling spatio-temporally continuous atmospheric motion. Likewise, directional information on weather changes is introduced from the entire historical series to mitigate estimation uncertainty and improve the performance of weather prediction. Extensive experiments on the Weather2K and Global Wind/Temp datasets demonstrate that CDPNet outperforms state-of-the-art models.