The Rhoynar had a huge impact on Dorne, which is why that is the most equal and liberal-minded part of Westeros, and almost zero impact outside of Dorne.

As for the rest, the ironborn culture prizes martial skills above all else and pushes women into subservient roles, actually rather moreso than in real Norse cultures (where women often had more say in things, particularly in defending home and hearth when their husbands/fathers/brothers were away). The Faith seems to have some influence in keeping women in lesser roles: whilst septas exist and are respected, they cannot become High Septon and I don't recall them being able to serve on the high council (the Devout) either. The maesters also refuse to accept women into their ranks (hence the whole Alleras/Sarella deception in AFFC), which tends to limit the opportunities for women to gain specialist knowledge in Westeros.

So I'd say the reasons for why women has less power than men in Westeros is the same as in the real medieval period - lack of eduction, lack of religious power and, with the confirmation of male-favouring inheritance (after the Dance of Dragons), little to no chance of a woman achieving political power and changing things. This may change when/if Daenerys takes the Iron Throne.