In the computer-aided diagnosis of lung cancer, the automatic segmentation of pulmonary nodules and the classification of benign and malignant tumors are two fundamental tasks. However, deep learning models often overlook the potential benefits of task correlations in improving their respective performances, as they are typically designed for a single task only. Therefore, we propose a multi-task network (MT-Net) that integrates shared backbone architecture and a prediction distillation structure for the simultaneous segmentation and classification of pulmonary nodules. The model comprises a coarse segmentation subnetwork (Coarse Seg-net), a cooperative classification subnetwork (Class-net), and a cooperative segmentation subnetwork (Fine Seg-net). Coarse Seg-net and Fine Seg-net share identical structure, where Coarse Seg-net provides prior location information for the subsequent Fine Seg-net and Class-net, thereby boosting pulmonary nodule segmentation and classification performance. We quantitatively and qualitatively analyzed the performance of the model by using the public dataset LIDC-IDRI. Our results show that the model achieves a Dice similarity coefficient (DI) index of 83.2% for pulmonary nodule segmentation, as well as an accuracy (ACC) of 91.9% for benign and malignant pulmonary nodule classification, which is competitive with other state-of-the-art methods. The experimental results demonstrate that the performance of pulmonary nodule segmentation and classification can be improved by a unified model that leverages the potential correlation between tasks.