Remember? It's a perfectly fine and useful word, is sexism.
Sexism is commonly considered to be discrimination and/or hatred against people based on their sex rather than their individual merits, but can also refer to any and all systemic differentiations based on the sex of the individuals.
Also see: "sexist," both noun and adjective.
Whereas patriarchy means, as I have understood it:
pa·tri·ar·chy (pā'trē-är'kē) pronunciation
n., pl. -chies. In both senses also called patriarchate.
1. A social system in which the father is the head of the family and men have authority over women and children.
2. A family, community, or society based on this system or governed by men.
Which, if you're gonna talk about oh say for example the "patriarchal" influences inherent in any society as derived from the Bible, esp. Old Testament, well and good: that makes sense, to me.
If you're gonna call oh i don't know the cultural pressure to wear lipstick "patriarchal," well, honestly, I'm not seeing it, so much. Sexist, arguably. Patriarchal? Only if you can point to a clear place where the "rule of the fathers" dictates the wearing of lipstick.
As I have understood it, traditional patriarchy, at least in this culture, tends rather to be against the whole face-painting business.
Whereas the face-painting itself has a number of origins, not least of which having to do with theatre and religious ritual and other things that really (in my opinion) have very little to do with the "rule of the fathers."
For that matter, personally I don't even see it as necessarily having to do with "discrimination based on sex."
Sure, it can be used that way; or at any rate the pressure for one sex to wear it and the other, not, certainly can be contrued that way.
But that doesn't mean it has to be that way, see.