I am by no means an expert in this field, but I do coach at a high level and have given lessons to many players over the years. Here are my thoughts on the two styles.
Rotational hitting is a very good style for those hitters who are strong enough to be power hitters. By "power hitters" I mean those girls who are strong enough to at least consistently hit the ball in the gaps with power, not just home run power. This style emphasizes the "L-position" at impact where the back elbow is kept tight to the body and the hands are inside the ball. Like the poster above said, this is more of a pull swing than a spray swing. The hips are supposed to begin to explode before the hands start to the ball and this causes some players to "pull off" the pitch and not reach the outside corner of the plate (especially on off speed stuff). Rotational swinging DOES generate more batspeed which is due in part to the violent hip action. It also leads to a few more strike outs because with all parts of the body in motion, it is easier to get "out of synch." It also leads to more backspin linedrives and fly balls than the other styles because of the tendency to uppercut. If the rotational swing is done perfectly, the player will keep the bat on the same plane from the time the bat enters the hitting zone until after contact. In reality, this style will more than likely have a slight uppercut swing path.
Linear hitting is more of a contact approach for those players that are not as physically strong. It emphasizes spraying the ball around the field and making contact. It can lead to overstriding if attention is not paid to the proper placement of the feet. With this method, the bat plane is more downward causing the player to hit down through the top of the ball and fewer popups. Most hitters that use linear hitting will hit a lot of topspin liners over the infield and ground balls. It is a very good approach for those players whose game is built around speed.
A few good points for each and a few bad points (just my opinion of course):
Rotational Good: Very little stride which leads to no change in eye level (easier to see the ball)
More batspeed which leads to more power
Hands stay inside the ball which leads to fewer balls yanked foul
Rotational Bad: With proper "L angle" at impact, it is easy to get jammed on inside pitches
Hips explode early leading to inability to reach off speed outside pitches
Linear Good: "Hands to the ball" approach leads to more spray hitting
More topspin is good for speedy hitters who rely on beating out grounders
Fewer strikeouts
Fewer fly balls and pop ups
Linear Bad: It can lead to overstriding
Less hip explosion leads to less power
There is a tendency to get "long" with this swing, players try to go out and get the
ball extending their arms and getting slow
When I begin to give lessons or work with our players, I try to use a combination of each approach. I also try to determine what kind of player I am working with. Obviously, a 85 lb 11th grader who runs a 2.5 home to first is not going to need to learn rotational hitting. She will want to hit the ball on the ground and use her speed to her advantage. Likewise, if the girl is bigger and stronger, I may want to teach her a more rotational approach so that she can use her power to her advantage.
All this is just my opinion of course and subject to rebuttal by those who are smarter than I am