Let \(P\) have dimension \(d\).
\(P\) is a simplex if it has exactly \(d+1\) vertices.
\(P\) is simple if every vertex lies on exactly \(d\) facets.
\(P\) is simplicial if every facet is a simplex.
The normal fan \(\Sigma_P\) is simplicial exactly when \(P\) is simple. Hence \(X_P\) is \(\QQ\)-factorial with at worst finite quotient singularities exactly when \(P\) is simple, and \(X_P\) is smooth exactly when in addition each vertex has its \(d\) edge directions forming a \(\ZZ\)-basis of \(M\).
(Each without the other)
The cube in \(\RR^3\) is simple and not simplicial: each vertex meets three facets, but the facets are squares. The octahedron in \(\RR^3\) is simplicial and not simple: each facet is a triangle, but each vertex meets four of them. The two are polar duals, which is the general picture — \(P\) is simple exactly when \(P^\circ\) is simplicial.
Explicitly, the cube \(P \subseteq \RR^3\) with vertices \((\pm 1, \pm 1, \pm 1)\) has facet normals \(\pm e_1, \pm e_2, \pm e_3\) and facet presentation \(\inp{m}{\pm e_i} \geq -1\). The origin is interior, so \(P^\circ\) is the octahedron with vertices \(\pm e_i\), and the maximal cones of the normal fan of \(P\) are the eight octants of \(\RR^3\), the cones over the facets of \(P^\circ\).
For a full-dimensional polytope \(P\), the vertices of \(P\) correspond to the maximal cones of \(\Sigma_P\): the cone at a vertex \(v\) is spanned by the inward facet normals of the facets containing \(v\). So \(v\) lies on exactly \(d\) facets if and only if its cone has exactly \(d\) rays, that is, if and only if the cone is simplicial. The facets of \(P\) correspond to the rays of \(\Sigma_P\), so the shape of a facet, such as whether it is a simplex, does not affect whether \(\Sigma_P\) is simplicial.