Rousseau declares that a woman should never, for a moment, feel herself independent, that she should be governed by fear to exercise her natural cunning, and made a coquettish slave in order to render her a more alluring object of desire, a sweeter companion to man, whenever he chooses to relax himself. (2.24)
Jean-Jacques Rousseau isn't going to win any awards for progressive gender politics. The guy pretty much thinks that women's only strength is their ability to manipulate and deceive men. He also thinks that their only purpose in life should be to look pretty and obey men.
[Women] were made to be loved, and must not aim at respect, lest they should be hunted out of society as masculine. (2.58)
Wollstonecraft knows that there's a big stigma against women who become well educated and try to gain power in male-dominated society. They'll get called manly or even the b-word just for demanding the same kind of respect that men do.
To preserve personal beauty, woman's glory! the limbs and faculties are cramped with worse than Chinese bands, and the sedentary life which they are condemned to live, whilst boys frolic in the open air. (3.12)
It is time to effect a revolution in female manners—time to restore to them their lost dignity—and make them, as a part of the human species, labour by reforming themselves to reform the world. (3.25)
Wollstonecraft thinks that the time has come for women to become equal members of society and to receive all of the same social privileges that men do. There is no way this type of change would ever make the world worse. It can only make things better because it will prepare women to contribute more productively to modern society.
I am fully persuaded that we should hear of none of these infantine airs, if girls were allowed to take sufficient exercise, and not confined in close rooms till their muscles are relaxed, and their powers of digestion destroyed. (4.34)
Wollstonecraft is certain that women would be much stronger and more active if they were given the same upbringing as men. She also thinks that women wouldn't act like children if they weren't treated like them.
This is the very point I aim at. I do not wish them [women] to have power over men; but over themselves. (4.34)
Wollstonecraft is not arguing for a woman-dominated society. What she wants is for women to be reasonable and intelligent, and there's no way for them to achieve this goal unless they receive the same education as men and learn to control their emotions by using their words.
In nurseries, and boarding-schools, I fear, girls are first spoiled; particularly in the latter. A number of girls sleep in the same room, and wash together. (7.23)
Wollstonecraft might be all about women's rights, but she strongly believes that having women hang out together all the time is a bad thing. She thinks that they should be in mixed company as often as possible, because they don't learn anything about the world when they only hang around themselves.
To say the truth women are, in general, too familiar with each other, which leads to that gross degree of familiarity that so frequently renders the marriage state unhappy. (7.24)
For Wollstonecraft, women spend too much time in the company of other women, and this familiarity ends up leading to unhappy marriages. Maybe it's because women just hang out all day and talk about their husbands' shortcomings.
But sense will always preponderate; and if women be not, in general, brought more on a level with men, some superiour women, like the Greek courtezans, will assemble the men of abilities around them, and draw from their families many citizens, who would have stayed at home had their wives more sense. (12.73)
If society doesn't step up and give women more rights and advantages, all of the smart women will find ways to attract men to them. And the people who'll be truly sorry will be the wives that these men leave behind so that they can be with someone smarter.
Let woman share the rights and she will emulate the virtues of man; for she must grow more perfect when emancipated, or justify the authority that chains such a weak being to her duty. (13.75)
There is nothing bad that could come from giving women more of a chance to educate themselves and to share the same civic rights that men have. If they get these rights and prove themselves unworthy, then all they'll do is justify the power that men have over them. Basically Wollstonecraft is saying "If you think women are lesser than men, let's prove it by giving them a chance. What are you afraid of?"