The shortest answer
A concept map is a network of concepts connected by labeled relationships. It may contain several hubs and cross-links. A mind map begins with one central topic and branches outward, usually without phrases explaining each connection.
Key differences
When to use a concept map
Choose a concept map when accuracy of relationships matters. It is especially useful for science systems, historical causes, clinical reasoning and any topic where one idea can connect to several parts of the structure.
When to use a mind map
Choose a mind map when you want to generate ideas quickly, collect associations around a topic or create a memorable one-page outline. The radial form is simple and fast.
A useful test
If you need to explain how or why ideas connect, choose a concept map. If you mainly need to list or brainstorm ideas around one topic, choose a mind map.