Algorithmic advantages of pathwidth over treewidth Treewidth plays an important role in FPT algorithms, in part because many problems are FPT parameterized by treewidth. A related, more restricted, notion is that of pathwidth. If a graph has pathwidth $k$, it also has treewidth at most $k$, while in the converse direction, treewidth $k$ only implies pathwidth at most $k\log n$ and this is tight.

Given the above, one may expect that there may be a significant algorithmic advantage to graphs of bounded pathwidth. However, it seems that most problems which are FPT for one parameter are FPT for the other. I'm curious to know of any counter-examples to this, that is, problems which are "easy" for pathwidth but "hard" for treewidth.

Let me mention that I was motivated to ask this question by running into a recent paper by Igor Razgon ("On OBDDs for CNFs of bounded treewidth", KR'14) which gave an example of a problem with a $2^{k}n$ solution when $k$ is the pathwidth and a (roughly) $n^k$ lower bound when $k$ is the treewidth. I am wondering if there exist other specimens with this behavior.

Summary: Are there any examples of natural problems which are W-hard parameterized by treewidth but FPT parameterized by pathwidth? More broadly, are there examples of problems whose complexity is known/believed to be much better when parameterized by pathwidth instead of 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