It is known that counting perfect matchings in a bipartite graph is #P-complete. On the other hand, finding a perfect matching belongs in P. Is there a problem, that exhibits the same behavior in parametrized complexity? (Decision in FPT and counting in #W or some other counting class?)
This is while Counting $k$-paths is $\#W$-hard.
A more interesting example (decision is even in $P$ while counting is parameterized-hard) would be counting $k$-matchings in bipartite graph. Not only that the problem is $\#P$-complete, it was recently shown to be $\#W$-hard as well !