Given how the supernatural defined their existence, character, and purpose that's hard for me to say. Just off the top of my head...
Buffy and Cordelia wouldn't have had the same life shaping experiences (and probably never met) so they'd remain the way they are for a long time. They'd also probably endure some harsh lessons even before they got out of their teen years. I'm not sure how that would affect them. Somehow, I could see Buffy, after awhile, getting into psychology to try to help others (and herself), but not being one of the best. She'd probably get depressed and give it up after awhile, or she'd become one of those jaded psychologists, and then marry the wrong man (but more or less coming out of it similar to how Joyce did with less denial).
Cordelia would get destroyed somehow pursuing an acting career in LA. (And I could see Harmony going to college but dropping out to get married, and then getting a lot in the divorce and living it up, almost forgetting Cordelia ever existed.)
Alternately, if Dawn still exists then maybe dealing with herself and a failure to "save" Buffy and/or her parents marriage might get her into psychology. She'd have a tendency to project her own issues into others, at least at first. And yet, perhaps because of the episode with the Gnarl (where no one can see Willow) makes me think of her as eventually becoming a forensic investigator, and perhaps shy about explaining that to guys, but also believing that the horrible crimes she investigates could never happen to her.
Willow is so mutable throughout the seasons that I can't really say about her either. She'd probably never go out with Oz but would probably get into computers, and perhaps develop a computer addiction.
Oz would've gotten through college, and he tests well. Perhaps he would've gotten into computers as well (since he was also talented) and perhaps he and Willow could meet that way (and maybe they could've dated and more then, but again Willow is so mutable that I don't know).
Tara's tale would be too sad to tell, and Anne's (just thought of her for some reason) probably not much better. Even dysfunctional (and perhaps alcoholic) Xander would pity her. I could see Xander being supported by Willow before Willow got tired of his crap.
Doyle would wind up similar to Xander, and I could see him committing suicide with many bitter regrets in his life (but again, hard to say as without the supernatural elements affecting his life, and who he was, then he, and his life, would be different).
Gunn is too reckless, I don't think he'd do well (and I could see him ending up in prison), though his sister might. Faith might also end up in prison, and not out of some noble attempt at redemption, either.
Giles and Wesley could do well in academia (though I could see Giles involved in some sort of sex scandal that might jeopardize that). Perhaps Fred could, too, it's hard to say given some other aspects of her character (but then those aspects I'm thinking of may never have become so exacerbated had she not been on Pylea for years).
And I have no idea about Angel, assuming he was born a mortal in the modern world. Angel Investigations would have to be completely different without the supernatural (from the visions to the problems), such as investigating insurance fraud and the like, working for people who are not helpless. But if he was more like he was as Liam then he'd become some disowned junkie with STDs living on the streets as best he can.
