I don't think that the weighted independent set problem in paths has many applications. However, as you point out yourself, it is a very good introduction to dynamic programming and to algorithmic graph theory; This is one of very many problems that are NP-complete on general graphs but become simpler (or indeed trivial) on restricted input instances.
When you know, and understand, the algorithm for (weighted) independent set in paths, it is not too hard to generalize this to (weighted) independent set in trees and even to graphs of bounded pathwidth and treewidth.
I think that this problem on paths (maybe even on trees, but here I'd love to be corrected) is purely educational. Indeed, I have hard times finding reasonable applications to Independent Set as well as Vertex Cover, Clique and Dominating Set.