Both mean the same thing in a broad sense, in that both mean you're persuading someone (a woman in this case). "Winning her over" might imply wooing a woman romantically (and successfully), though not necessarily: you can win someone over in a business deal, for example. The only other clear distinction that occurs to me is that "to win someone over" is more idiomatic, while "to persuade" is more general. Still, they're both extremely common verbs.