In recent years, networks with higher-order interactions have emerged as a powerful tool to model complex systems. Comparing these higher-order systems remains however a challenge. Traditional similarity measures designed for pairwise networks fail indeed to capture salient features of hypergraphs, hence potentially neglecting important information. To address this issue, here we introduce two novel measures, Hyper NetSimile and Hyperedge Portrait Divergence, specifically designed for comparing hypergraphs. These measures take explicitly into account the properties of multi-node interactions, using complementary approaches. They are defined for any arbitrary pair of hypergraphs, of potentially different sizes, thus being widely applicable. We illustrate the effectivenes of these metrics through clustering experiments on synthetic and empirical higher-order networks, showing their ability to correctly group hypergraphs generated by different models and to distinguish real-world systems coming from different contexts. Our results highlight the advantages of using higher-order dissimilarity measures over traditional pairwise representations in capturing the full structural complexity of the systems considered.