Hyperspectral images (HSIs) are prone to noise because of the imaging mechanism and environment. This article proposes a multitask sparse representation (MTSR) modelinspired neural network for HSI denoising. Unlike other deep learning (DL)-based methods, our network is interpretable, whose network architecture is induced by unfolding the iterative optimization of an MTSR model. On one hand, the model globally represents the common structure among bands, such as image edges, with the shared sparse coefficients. On the other hand, it separately encodes the unique structure of individual bands with unshared ones to capture image details. Accordingly, our network has three modules: the shared sparse representation (SSR) module, the unshared sparse representation (USR), and the image reconstruction (IR) module. All the modules are connected with a specific operation of the iterative optimization algorithm, equipping the network with clear physical interpretation. Experimental results on both the synthetic and real-world datasets demonstrate the superior performance of our method, visually and quantitatively. The codes will be publicly available at https://github.com/bearshng/mtsrnn for reproducible research.