Skin cancer poses a serious global health challenge, where timely and precise diagnosis is essential to improve patient outcomes. Recently, neural networks have proven to be highly effective tools for automated skin cancer classification, significantly advancing the field of dermatology. This paper introduces a novel approach to generate edge maps from dermoscopic images using a holistically nested edge detector model. These edge maps enhance the detection of shape and symmetry irregularities, which are key indicators of malignancy, and improve the focus on relevant regions of interest. We then propose an edge-guided dual-branch neural network, called EDB-Net, for the classification task. Branch 1 handles edge maps, while Branch 2 processes original dermoscopic images. To highlight significant regions and focus on specific lesion areas, we incorporate a novel channel-spatial synergistic attention block within Branch 2. Additionally, we introduce a unique strategy to modulate the generated attention maps using edge features extracted from the edge maps in Branch 1, creating edge-guided features that refine the overall feature representation. In the final stage, both edge-guided and attention-aided features are combined, producing more distinct and contextually relevant outputs, thereby significantly enhancing classification performance. Our model achieves accuracies of 0.927 and 0.848 on the challenging HAM10000 and ISIC 2016 datasets, respectively, without employing any data augmentation. The source code of the proposed model is available at: https://github.com/Cmatermedicalimageanalysis/EDB_Net .