One thing before I start: I am not, nor have I ever been, a girl. Therefore what follows is a mixture of fantasy, postulation and an unreconstructed male perspective. For my first-hand experience of my side of the story, see Boys' Sleepovers.
Girls' sleepovers exist for these reasons and these reasons only:
Getting moist-pantied over romantic leads
Grease. Dirty Dancing. Shakespeare In Love. Teenage girls are even worse than teenage boys for getting hysterical with lust for celebrities. No sleepover worth its salt will pass without a film featuring a hunk and a lot of weeping.
Learning about makeup and cheap white wine
There's got to be a reason why, seemingly overnight, the girls in your class suddenly appeared at school looking about 5 years older and as hot as hell. Getting giggly on Lambrusco Bianco is a pleasant side-effect, especially at later intergender sleepovers.
Having very serious discussions about which guys they'd go with and how far they'd go
With guys, it's simple. 'Which girl would you have sex with?' The question may expand to location, position and colour of underwear she would choose, but essentially that would be the discussion. With girls, the questions are 'Do you like him enough to let him french kiss you?'; 'Which guy would you let see you naked?'; 'If you were going steady, would you let him touch you...down there?'. This conversation can last for years. Meanwhile, the guys have agreed that Becky Blackwell would be best in a yellow lace thong and matching push-up bra, doggy-style on the kitchen floor, and moved on to talk about how to complete Street Fighter II.
Pillow fights in pyjama tops and panties
Indulge me please. There's no more perfect image to me than the idea that the girls I grew up with had sleepovers whereby they pampered each other while watching a chick flick and eating popcorn, got tipsy on smuggled-in liquor, spent the entire night wearing nothing more than tight t-shirts or checked pyjama tops and their underwear, had giggly pillow fights and all fell asleep cuddled together. If my illusions are shattered, I fear I have nothing left to live for.
Update: It has been pointed out to me that another reason for teenage girls to have sleepovers is lesbian experimentation. Now while the mere thought of such nubile teen loveliness fills my heart (and other parts) with joy, it is a)clearly a bridge too far, even further than my panty-flashing pillow fighting dreams, and b)too painful for me to consider given that the girls I grew up with about as hot as jailbait ever got.