Problems that are easy on unweighted graphs, but hard for weighted graphs
In the world of approximation algorithms there is the capacitated vertex cover problem.
Given $G=(V,E)$ and integer capacities $c(v)$ for each $v \in V$ the goal is to find
a minimum sized vertex cover for $G$ where the number of edges covered by $v$ is at most $c(v)$. This problem has a constant factor approximation in the unweighted case (that is, we want to minimize the size of the vertex cover) while it is $\Omega(\log n)$-hard (unless $P = NP$) in the weighted case (each vertex has a weight $w(v)$ and we want to minimize the weight of the cover).