How are the shortest-path and travelling-salesman problems given above similar? How are they different?

They are similar in the sense that both traverses a graph and tries to find out the shortest path with minimum cost (sum of the weights).

They are different because **shortest-path problem** finds a path *between two points* such that sum of the weights is minimized. Whereas, **travelling-salesman problem** finds the path *covering all the points* (start and end point is same) such that sum of the weights is minimized. Also, shortest-path problem is P complex and travelling-salesman is NP-complete.