In signaling games, it seems that the Cho Kreps refinement (intuitive criterion) is the go to refinement for eliminating bad sequential equilibria. Divine equilibrium and perfect sequential equilibrium are also interesting refinements though. It seems more common that these are compared to Cho Kreps. Are there any interesting examples comparing the differences between perfect sequential and divine equilibrium? I've never seen the two concepts discussed together.
In Cho, I. K., & Kreps, D. M. (1987). Signaling games and stable equilibria. The Quarterly Journal of Economics, 179-221, Banks & Sobel's Divine (and Universally Divine) equilibrium concepts are presented in Section IV.4 as a stand-alone concept. On the other hand Grossman & Perry's Perfect Sequential Equilibrium concept is just mentioned in Section IV.5 which has the title "Never a weak best response".
In the Banks, J. S., & Sobel, J. (1987). Equilibrium selection in signaling games. Econometrica, 647-661. paper on Divine Equilibrium, page 654 (near the end of Section 3), we read "(...) This condition is more restrictive than universal divinity because (...)" , "this condition" being "never a weak best response".
So it appears that Perfect Sequential Equilibrium (PSE) is a stronger equilibrium filtering criterion than Divine Equilibrium. This accords with
Theorem 2 of Banks & Sobel : Every signaling game has a divine equilibrium
to be contrasted with Section 4. of the Grossman, S. J., & Perry, M. (1986). Perfect sequential equilibrium. Journal of economic theory, 39(1), 97-119. paper introducing PSE, where they show by means of an example that a Perfect Sequential Equilibrium may fail to exit.
A paper that applies both concepts is Beggs, A. W. (1992). The licensing of patents under asymmetric information. International Journal of Industrial Organization, 10(2), 171-191.. In Section 3.2 a result is derived by an appeal to the PSE concept. Then the authors note that, given an additional condition, they could obtain the same result by an appeal to Divinity. This agains shows that PSE, when it exists, is stronger than Divine equilibrium. Here too an example is offered for a case when the PSE does not exist.