A sequence of r.v.s $z_1, z_2, \ldots$ is said to converge in distribution to a r.v. $z$ if $$\lim_{N\rightarrow \infty} P_{z_N}(z) = P_z(z)$$ for all z at which $P_Z$ is continuous.
A sequence of r.v.s $z_1, z_2, \ldots$ is said to converge in divergence to a r.v. $z$ if $$\lim_{N\rightarrow \infty} D(P_{z_N}||P_z) = 0$$
Intuitively, why does convergence in divergence give us more than convergence in distribution? If each individual value that the distribution takes converges, then it seems like (hand-wavy / intuitively) that the $\log \frac{p_{z_n}(z)}{p_z(z)}$ in the divergence would go to 0 for all z as well?
Convergence in distribution pays attention to the topology of the underlying spaces, so for instance, as $h\to0$, the point mass $\delta_h$ concentrated at $h$ converges in distribution to the point mass $\delta_0$ at $0$. But the divergence does not: it only looks at the values of RN derivatives of the distributions involved. Intuitively, in terms of the distributions' density functions, the one sees how close they are vertically and horizontally, and the other only how close they are vertically.