Accurately segmenting brain tumors is vital for diagnosing and treating the disease. Recently, convolutional neural networks (CNNs) have obtained notable performances in segmenting brain tumors. Nevertheless, they have limited capability to utilize long-range (or global) dependencies. In contrast, Transformers can successfully model global dependencies but cannot adequately capture local dependencies. It is crucial to utilize both local and global dependencies to perform accurate brain tumor segmentation. Consequently, several studies have attempted to combine the benefits of CNN and Transformer. However, effectively capturing local and global information remains challenging. Thus, we introduce a new 3D U-Net variant termed Dual-Branch Transformer-CNN Network (DBTC-Net), in which the encoder contains two branches built using Swin Transformer and CNN to effectively utilize global and local information. Furthermore, we design a Swin Transformer Channel Attention (SwinTCA) block by modifying the Swin Transformer blocks to enable it to capture location-wise channel dependencies in addition to global spatial dependencies. Moreover, we propose a Transformer Convolution Feature Combination (TCFC) block that effectively combines the complementary global and local features from the Transformer and CNN encoder blocks to improve the feature representation capability. In addition, a Multi-Scale Context Combination (MCC) block is introduced in the bottleneck to handle the variations in tumor size by utilizing multi-scale contextual features and forcing the network to concentrate on the tumor area. Extensive experimentation on the BraTS 2021 and 2020 benchmark datasets proves the success of the introduced components. The results reveal that DBTC-Net outperformed the CNN- and Transformer-based state-of-the-art networks.