No lawyer, but .....
I think the difference is that the salary cap in the Saracens case was applied equally across all clubs in the league. The APT rules, as can be shown by the timing of their introduction and the definition of associated party, unfairly affect clubs that have a certain type of owner and a consequent type of business model.
I don't know if the club really has the goal of having the whole set of APT rules thrown out (if they wanted that they could have tried that between November 2023 and February 2024, for example). I suppose they could, though, and if they do they will probably not be successful for the stated reasons (which are probably also the reasons FFP itself has never been challenged).
I think it is more likely the club are aiming to have the more discriminatory rules and definitions changed to more generic terms that apply equally irrespective of who the owners care and what the business model is. They could have success there, I think (and let's not forget there were rules around the value of related party transactions before the APT rules, it's not like there isn't a fall-back position for the PL).
Which may be why some press reports say some wins and some defeats for both parties.
By the way, this doesn't mean that the club isn't asking for an assessment of the whole rule set. Why not aim high? I am suggesting their goal is to remove the discriminatory and unnecessarily burdensome nature of the February rule changes to the club and other clubs in the same situation.