Quantifying structural similarity between complex networks presents a fundamental and formidable challenge in network science, which plays a crucial role in various fields, such as bioinformatics, social science, and economics, and serves as an effective method for network classification, temporal network evolution, network generated model evaluation, etc. Traditional network comparison methods often rely on simplistic structural properties such as node degree and network distance. However, these methods only consider the local or global aspect of a network, leading to inaccuracies in network similarity assessments. In this study, we introduce a network similarity comparison method based on the high-order structure. This innovative approach takes into account the global and the local structure of a network, resulting in a more comprehensive and accurate quantification of the network difference. Specifically, we construct distributions of higher-order clustering coefficient and distance between nodes in a network. The Jensen-Shannon divergence, based on these two distributions, is used to quantitatively measure the similarity between two networks, offering a more refined and robust measure of network similarity. To validate the effectiveness of our proposed method, we conduct a series of comprehensive experiments on the artificial and the real-world network, spanning various domains and applications. By meticulously fine-tuning the parameters related to three different artificial network generation models, we systematically compare the performances of our method under various parameter settings in the same network. In addition, we generate four different network models with varying levels of randomization, creating a diverse set of test cases to evaluate the robustness and adaptability of the method. In artificial networks, we rigorously compare our proposed method with other baseline techniques, consistently demonstrating its superior accuracy and stability through experimental results; in real networks, we select datasets from diverse domains and confirm the reliability of our method by conducting extensive similarity assessments between real networks and their perturbed reconstructed counterparts. Furthermore, in real networks, the rigorous comparison between our method and null models underscores its robustness and stability across a broad spectrum of scenarios and applications. Finally, a meticulous sensitivity analysis of the parameters reveals that our method exhibits remarkable performance consistency across networks of different types, scales, and complexities.