So, what is the difference between maximum flip and maximum flare ?
Lot of variables but maximum flip often occurs with less flare because when a ball is flaring more fresh surface of the ball in hitting the lane which is generating friction and slowing down the ball, bleeding off rotational energy and depending hooking as well. When a ball stores more of its energy for the back half of the lane it has more energy to turn left at the end when it exits the oil, sees the dry friction and generates maximum flip. Maximum hook generally occurs with more flare (up to a point) which starts earlier and is more even. Lot of other variables in involved but 10000 foot view.
(edit: maximum flare doesn't necessary mean maximum hook depending on ball, surface, style and conditions as its complicated)