In the show, the falling out between Tully and Kate actually happens when Tully is looking after Marah while the teenager is grounded. She lets Marah go out to meet a friend, but Marah calls her in a panic after the two end up at a frat party.
After picking Marah up, the two get into a car accident, and Kate vows to never forgive Tully for allowing Marah out of the house. Marah later tricks Tully into coming over to reconcile with Kate, but the latter doesn’t accept her apology.
Source: https://www.distractify.com/p/what-did-tully-do-to-kate-in-firefly-lane