Deep learning methods have made some achievements in the automatic skin lesion recognition, but there are still some problems such as limited training samples, too complicated network structure, and expensive computational costs. Considering the inherent power-efficiency, biological plausibility and good image recognition performance of spiking neural networks (SNNs), in this paper we make malignant melanoma and benign melanocytic nevi skin lesions classification using convolutional SNNs with unsupervised spike-timing-dependent plasticity (STDP) learning rule. Efficient temporal coding, event driven learning rule and winner-take-all (WTA) mechanism together ensure sparse spike coding and efficient learning of our networks which achieve an average accuracy of 83.8%. We further propose to use feature selection to select more diagnostic features to improve the classification performance of our networks. Our SNNs with feature selection reach an average accuracy of 87.7%. Experimental results show that comparing to CNNs that need to be trained from scratch, our SNNs (with and without feature selection) not only achieve much better classification accuracies but also have much better runtime efficiency. Moreover, although the pretrained CNNs models can achieve similar running time, our proposed SNNs are more stable and easier to use than the pretrained CNNs because we do not need to try many pretrained models any more, and our SNNs also have much better classification accuracies than the pretrained CNNs. In addition, our networks have only three convolutional layers, and the complexity of the model and the parameters that need to be trained in the networks are greatly reduced. Our works show that STDP-based SNNs are very beneficial for the implementation of automated skin lesion classifiers on small portable devices.