Skin cancer is one of the most serious threats to human health among skin lesions. Computer-aided diagnosis methods can assist patients in identifying and detecting skin lesion types early, thereby enabling corresponding treatments. In this paper, we propose a dual-branch neural network model Conformer with Residual Cosine Similarity Attention and Bidirectional Convolutional fusion strategy, named RCSABC-Conformer. The core of this network structure comprises three parts: a Convolutional Neural Network (CNN) branch with Residual Cosine Similarity Attention (RCSA), a Transformer branch, and a Feature Couple Unit with Bidirectional Convolutional strategy (BC-FCU). The RCSA module calculates the cosine similarity value between the feature map generated by the convolutional operation and the feature map of the residual edge to assess whether their semantic information is similar. The semantic information of similar parts is weighted by exponential normalization to enhance the network's memory of similar features of the same type of skin lesion. The BC-FCU module interactively fuses local features and global representations of skin lesion images with different resolutions in the two branches. Specifically, when the global representations is integrated into local features, we introduce a new bidirectional convolution strategy to extract the feature map from both forward and backward directions, and then select the element with the smaller feature value from the two directions to fuse into local features. In this way, we can minimize the interference of the artifact features extracted by the Transformer branch on the CNN branch. In addition, taking advantage of the Transformer branch's capacity to construct global representations, our model can learn contextual semantic information of normal skin and lesion areas to enhance model robustness. We conducted experiments on three datasets, consisting of clinical and dermoscopic skin lesion images, as well as a hybrid of both. The experimental results show that RCSABC-Conformer outperforms both advanced and classical classification methods in terms of classification accuracy across all three datasets, without requiring an increase in the number of parameters and computational complexity. Compared with the baseline model, the classification accuracy of our proposed method improves by 2.40%, 5.39%, and 4.44% on the three datasets, respectively. To the best of our knowledge, this is the first study to apply an interactive fusion dual-branch network for multi-disease classification on different modalities of skin lesion databases. Code will be available at https://github.com/AlenLi817/RCSABC-Conformer.