Nothing. Or let’s be precise: not much. Both are things (nodes, vertices) connected by other things (links, edges) and any distinction is pretty much nothing but tradition. My PhD dissertation has a network science part. When I started writing about Erdős-Rényi graphs in this part I realized I had to clear up the relationship between the networks and graphs. After some research I tentatively wrote “Networks are essentially graphs”. Unfortunately, this statement did not resound well with one of my opponents, so I had to dig deeper.