There is at least one pair of examples of diffeomorphic but not deformation equivalent three-dimensional Calabi-Yau manifold (Ruan-Gross). The example is explained on pages 47-48 of this paper: http://arxiv.org/pdf/math/9806111.pdf

Otherwise it is of course natural to try to distinguish Calabi Yau three folds by their diffo type. Note that in dimension six two smooth compact manifolds that are homeomorphic are necessarily diffeomorphic, so the classifications up to homeo and diffeo are the same.
Classification of simply connected 6-manifolds with torsion free homology according to diffeo is given by a theorem of Wall (the essential bit here is the cubic intersection form on $H^2(M^6,\mathbb Z)$). I am not aware of (current) classification work in this direction for Calabi-Yau 3-fold. But I think someone who would like to do this should use computer (the majority of examples of CY 3-folds are an outcome of a certain computer program). And it seems to me that it should be possible in principle to improve the existing algorithm so that it computes not only betti numbers, but also multiplication on $H^*$ and so the type as well.

Concerning the topology of CY 3-folds, on can say at least that the fundamental group of CY manifolds is finite (or, depending on definition of what you call CY manifold, virtually Abelian). At the same time general Kahler manifolds can have very sophisticated fundamental groups.