Which graph problems are $W[1]$-Hard on directed(/weighted) graphs but FPT on undirected(/unweighted) graphs?
The disjoint paths problem: given $G$ and $k$ pairs of nodes, are there node disjoint paths connecting the given pairs. Parameterized by $k$, in FPT when $G$ is undirected from the seminal work of Robertson and Seymour. NP-Hard for $k=2$ when $G$ is directed - from work of Fortune, Hopcroft and Wylie (1980).