Removing adverse weather conditions like rain, fog, and snow from images is a challenging problem. Although the current recovery algorithms targeting a specific condition have made impressive progress, it is not flexible enough to deal with various degradation types. We propose an efficient and compact image restoration network named DAN-Net (Degradation-Adaptive Neural Network) to address this problem, which consists of multiple compact expert networks with one adaptive gated neural. A single expert network efficiently addresses specific degradation in nasty winter scenes relying on the compact architecture and three novel components. Based on the Mixture of Experts strategy, DAN-Net captures degradation information from each input image to adaptively modulate the outputs of task-specific expert networks to remove various adverse winter weather conditions. Specifically, it adopts a lightweight Adaptive Gated Neural Network to estimate gated attention maps of the input image, while different task-specific experts with the same topology are jointly dispatched to process the degraded image. Such novel image restoration pipeline handles different types of severe weather scenes effectively and efficiently. It also enjoys the benefit of coordinate boosting in which the whole network outperforms each expert trained without coordination. Extensive experiments demonstrate that the presented manner outperforms the state-of-the-art single-task methods on image quality and has better inference efficiency. Furthermore, we have collected the first real-world winter scenes dataset to evaluate winter image restoration methods, which contains various hazy and snowy images snapped in winter. Both the dataset and source code will be publicly available.