Algorithmic advantages of pathwidth over treewidth
It is shown that [1] the Mixed Chinese Postman Problem (MCPP) parameterized by pathwidth is $W[1]$-hard, even if all edges and arcs of the input graph $G$ have weight $1$ and is FPT with respect to treedepth. This is the first problem are aware of that has been shown to be $W[1]$-hard with respect to treewidth but FPT with respect to treedepth. Note that the pathwidth of a graph lies between its treewidth and treedepth.

The Steiner Multicut problem, which asks, given an undirected graph $G$, a collection $T = \{T_1, . . . , T_t\}$, $T_i ⊆ V(G)$, of terminal sets of size at most $p$, and an integer $k$, whether there is a set $S$ of at most $k$ edges or nodes such that of each set $T_i$ at least one pair of terminals is in different connected components of $G \ S$.

Node Steiner Multicut, Edge Steiner Multicut, and Restr. Node Steiner Multicut are $W[1]$-hard for the parameter $k$, even if $p = 3$ and $tw(G) = 2$ [2]. 

[1] https://core.ac.uk/download/pdf/77298274.pdf

[2] http://drops.dagstuhl.de/opus/volltexte/2015/4911/pdf/11.pdf