No, the authors are not suggesting that the two manifolds are homeomorphic and not diffeomorphic.
Your two manifolds are Riemannian manifolds: smooth manifolds equipped with a Riemannian metric. The volume of a Riemannian manifold is preserved under isometry: it's defined by integrating the volume form over the manifold, and an isometry $M \to N$ pulls back the volume form of $N$ to the volume form of $M$. But this is far from true just for diffeomorphisms.
For instance, the $2$-sphere of any radius is diffeomorphic to the 2-sphere of any other radius; but the volume of $S^2(r)$ is $4\pi r^2$. So we can give a sphere a Riemannian metric of any given volume.
The authors in your paper show that the two Riemannian manifolds are not isometric by showing that they have different volumes; but they are still diffeomorphic.
