In the books: Mia and Lilly come to the blow of all blows. Their eventual falling out is because Mia dumps Michael, Lilly's brother, and then eventually starts dating J.P., Lilly's ex. Again, not the most pleasant chunk of the series, but Mia def needed some space. Mia's time away from Lilly allowed for her to find her own voice and express her own thoughts without Lilly's judgement. They do eventually reconcile, though, and they're better people because of it.
In the movie: Lilly gets mad at Mia for one, her glow-up and two, not appearing on her cable show Shut Up and Listen. Fight-worthy? I think not. And they make up in like five seconds and no lessons are learned from the ordeal.