Segmentation and classification of breast tumors are two critical tasks since they provide significant information for computer-aided breast cancer diagnosis. Combining these tasks leverages their intrinsic relevance to enhance performance, but the variability and complexity of tumor characteristics remain challenging. We propose a novel multi-task deep learning network (NMTNet) for the joint segmentation and classification of breast tumors, which is based on a convolutional neural network (CNN) and U-shaped architecture. It mainly comprises a shared encoder, a multi-scale fusion channel refinement (MFCR) module, a segmentation branch, and a classification branch. First, ResNet18 is used as the backbone network in the encoding part to enhance the feature representation capability. Then, the MFCR module is introduced to enrich the feature depth and diversity. Besides, the segmentation branch combines a lesion region enhancement (LRE) module between the encoder and decoder parts, aiming to capture more detailed texture and edge information of irregular tumors to improve segmentation accuracy. The classification branch incorporates a fine-grained classifier that reuses valuable segmentation information to discriminate between benign and malignant tumors. The proposed NMTNet is evaluated on both ultrasound and magnetic resonance imaging datasets. It achieves segmentation dice scores of 90.30% and 91.50%, and Jaccard indices of 84.70% and 88.10% for each dataset, respectively. And the classification accuracy scores are 87.50% and 99.64% for the corresponding datasets, respectively. Experimental results demonstrate the superiority of NMTNet over state-of-the-art methods on breast tumor segmentation and classification tasks.