If the fibers are Kähler manifolds, then the Hodge numbers of the fibers are constant and hence by Grauert's theorem, the direct images of the relative Hodge bundles are locally free (i.e., vector bundles). In other words, in this case $R^q\pi_*\Omega_W^p$ is a vector bundle for every $p,q\in \mathbb N$.

If the fibers are compact Kähler manifolds, then the Hodge numbers of the fibers are constant and hence by Grauert's theorem, the direct images of the relative Hodge bundles are locally free (i.e., vector bundles). In other words, in this case $R^q\pi_*\Omega_W^p$ is a vector bundle for every $p,q\in \mathbb N$.