How are the shortest path above and the traveling salesperson problems given above similar?
How are they different?

Both problems are graph problems and seek the optimal path to fulfill certain criteria.
The difference is that while a Traveling Sales Person is seeking to tour all the nodes, the shortest path is seeking to tour a few nodes optimally.
In the shortest path, when represented as a graph the objective is to find the smallest weight along a path between the two nodes.
In traveling sales person the object is to minimize the total distance traveled in a tour visiting each node exactly once