The medical sciences are facing a major problem with the auto-detection of disease due to the fast growth in population density. Intelligent systems assist medical professionals in early disease detection and also help to provide consistent treatment that reduces the mortality rate. Skin cancer is considered to be the deadliest and most severe kind of cancer. Medical professionals utilize dermoscopy images to make a manual diagnosis of skin cancer. This method is labor-intensive and time-consuming and demands a considerable level of expertise. Automated detection methods are necessary for the early detection of skin cancer. The occurrence of hair and air bubbles in dermoscopic images affects the diagnosis of skin cancer. This research aims to classify eight different types of skin cancer, namely actinic keratosis (AKs), dermatofibroma (DFa), melanoma (MELa), basal cell carcinoma (BCCa), squamous cell carcinoma (SCCa), melanocytic nevus (MNi), vascular lesion (VASn), and benign keratosis (BKs). In this study, we propose SNC_Net, which integrates features derived from dermoscopic images through deep learning (DL) models and handcrafted (HC) feature extraction methods with the aim of improving the performance of the classifier. A convolutional neural network (CNN) is employed for classification. Dermoscopy images from the publicly accessible ISIC 2019 dataset for skin cancer detection is utilized to train and validate the model. The performance of the proposed model is compared with four baseline models, namely EfficientNetB0 (B1), MobileNetV2 (B2), DenseNet-121 (B3), and ResNet-101 (B4), and six state-of-the-art (SOTA) classifiers. With an accuracy of 97.81%, a precision of 98.31%, a recall of 97.89%, and an F1 score of 98.10%, the proposed model outperformed the SOTA classifiers as well as the four baseline models. Moreover, an Ablation study is also performed on the proposed method to validate its performance. The proposed method therefore assists dermatologists and other medical professionals in early skin cancer detection.