Role is a fundamental concept in the analysis of the behavior and function of interacting entities represented by network data. Role discovery is the task of uncovering hidden roles. Node roles are commonly defined in terms of equivalence classes, where two nodes have the same role if they fall within the same equivalence class. Automorphic equivalence, where two nodes are equivalent when they can swap their labels to form an isomorphic graph, captures this common notion of role. The binary concept of equivalence is too restrictive and nodes in real-world networks rarely belong to the same equivalence class. Instead, a relaxed definition in terms of similarity or distance is commonly used to compute the degree to which two nodes are equivalent. In this paper, we propose a novel distance metric called automorphic distance, which measures how far two nodes are of being automorphically equivalent. We also study its application to node embedding, showing how our metric can be used to generate vector representations of nodes preserving their roles for data visualization and machine learning. Our experiments confirm that the proposed metric outperforms the RoleSim automorphic equivalence-based metric in the generation of node embeddings for different networks.