The main in-your-face advantages of WHM over AST are Benediction and Assize. Benediction is strong; especially if used with Living Dead, WHM can do a lot of DPS with Living Dead whereas it requires a legitimate amount of MP/resources/time to save a DRK on AST. It can be used for pure DPS or a big MP saver or just a big emergency heal, just a great skill overall that's unique to WHM and is one of the reasons WHM can DPS more often than AST. Assize is for instant MP occasionally, DPS on adds/AoE, or AoE burst healing, all strong attributes AST doesn't really get on demand. Most of the kit is otherwise sidegrade-ish, but notably Disable being undisputed better than regular Virus in current fights, with how Anti-Body works.
Asylum is better in some scenarios, Collective is better in some scenarios (usually Collective can be better for mega attacks but if it's used as just a tank regen it's usually better to plop down Asylum than run up, possibly in the way of mechanics/cleaves, and wait for 2-3s for Collective to tick), same with DS/Synastry (DS usually better but when Synastry is hot it's HOT) and Tetra/Essential and PoM/Lightspeed (although in general, in current content, PoM is more useful since content has so little mobility required). Arguably Shroud's aggro control over LA is a real advantage, and it is, but that's more an annoyance than a strong advantage because the two points aggro can be an issue, A3 and Thordan, the boss isn't threatening for a while past the phase shift.
I'm not much for saying AST is weaker, and that's not the point of this post, in fact I'm kind of an AST white knight these days. To answer your question, Assize and Benediction are just things AST doesn't have an active answer to besides making sure they're not needed. Usually with more time spent healing or making sure there are less mistakes to warrant them.