I want to add to Gozz's answer. Spanning trees are unrooted undirected trees. Shortest path trees are directed and rooted by construction, and only need to cover nodes that are reachable from the source node. The directed version of a spanning tree is an arborescence. Finding a spanning arboresence is much more complicated than finding a spanning tree: [see here](https://en.wikipedia.org/wiki/Edmonds%27_algorithm). The point is that spanning trees only work on undirected graphs, and shortest path trees work on both directed and undirected graphs. If you run Dijkstra's on an undirected graph, then they're still not always the same thing as Gozz points out.