We carry out a critical assessment of machine learning and deep learning models for the classification of skin tumors. Machine learning (ML) algorithms tested in this work include logistic regression, linear discriminant analysis, k-nearest neighbors classifier, decision tree classifier and Gaussian naive Bayes, while deep learning (DL) models employed are either based on a custom Convolutional Neural Network model, or leverage transfer learning via the use of pre-trained models (VGG16, Xception and ResNet50). We find that DL models, with accuracies up to 0.88, all outperform ML models. ML models exhibit accuracies below 0.72, which can be increased to up to 0.75 with ensemble learning. To further assess the performance of DL models, we test them on a larger and more imbalanced dataset. Metrics, such as the F-score and accuracy, indicate that, after fine-tuning, pre-trained models perform extremely well for skin tumor classification. This is most notably the case for VGG16, which exhibits an F-score of 0.88 and an accuracy of 0.88 on the smaller database, and metrics of 0.70 and 0.88, respectively, on the larger database.