Generally speaking
“Lawful” = following the rules to get the desired result. The process guarantees the outcome.
“Chaotic” = Achieving the goal is more important than the rules. Do whatever it takes.

For LN and CN the quarrel isn’t about the goals it is about the right way to achieve the goals.
Chaotic gods will mess with Lawfuls by creating situations where following the process leads to undesirable outcomes.
Lawful gods will mess with Chaotics by creating situations where the unintended consequences of not following the rules is undesirable.

Where both gods are of the same G/N/E alignment this will be more about who gets more credit for desirable outcomes and therefore more followers and ultimately more souls. Both want N to win