The majority of Ashkenazic communities hail from (majority-Christian) Eastern Europe, while most Sephardic communities come from the (majority-Muslim) Middle East and North Africa.
Islam seems to take a rather strict view on the role of women. According to a strict interpretation (such as is followed in modern-day Saudi Arabia), women are not allowed to leave the house on their own, must be (fully) covered, and are generally required to submit to their "dominant male".
Christianity gives women some more free reign. From https://en.wikipedia.org/wiki/Women_in_Christianity#Women_in_church_history, it appears that while women historically were not appointed to high positions, they were certainly considered people in their own right, and did not have the restrictions mentioned above. See also What is the source for Rav Soloveitchik's assertion that 1st Century Christians introduced mixed seating for prayers?
Now that we've clarified some of the surrounding culture, let's move on to the Jewish sources.
The classic position held by Ashkenazim is that even if women are exempt, they are still allowed to make the bracha. Sefardic poskim hold that she should not make the bracha. The classic source for this is SA OC 17:2, where the Rama has to point out that women should say the bracha on tzitzit.
(Interestingly, this position seems to be sort of reversed when it comes to women wearing tefillin - Ashkenazim discourage them from wearing it (while still allowing the bracha if she does), while Sephardim don't discourage it (but also without allowing a bracha).)
I don't have a source for this, but i have heard that while Ashkenazi women are supposed to follow the standard text of davening, Sfardi women just make up a prayer from the heart once a day.
Based on the two things i just mentioned, and there are probably more, it seems that Ashkenazi women take a bigger role in communal/religious life than Sfardi women.
Are there any sources that indicate that the positions on women were influenced by the surrounding culture?