Can a shortest-path tree be a also maximum spanning tree?
Let A-B-C be connected graph with edges like shown (AB and BC) the shortest path (the only possible) from A to C is through B. To include all vertices in the maximum spanning tree all vertices are needed, so this is small example (the smaller, A-B is also valid).

Maximum spanning tree have maximal weights, but may be not unique for given graph. The shortest path tree (which I assume is the weighted one) have path weights but also is not guaranteed to be unique.

From the definitions finding graph with cycles will be harder - taking minimal example, a full graph with vertices D, E, F and distances |DE| = 3, |EF| = 2, |DF| = 3, the shortest path from E to F is 2, but the maximal spanning tree will not contain this edge.
The two structures may overlap when there are no cycles or weights are equal, otherwise they will get opposite edges when possible.