Patriarchy is, according to a certain contingent of Internet feminism, everywhere. It's like midichlorians; it's in the air and in every word we speak and it dwells within every human heart. You get up in the morning in a patriarchy bed and brush your teeth with patriarchy, then put on some patriarchy and head out the door to patriarch all day. And you can do all this without even knowing it.
Whereas sexism is a thing. Sexism is a specific action or attitude that people can have in different degrees. Maybe everyone's a little sexist sometimes, but the amount of sexism differs tremendously from "sometimes thinks unfair things but tries to correct self" to "women exist to destroy men." Sexism is detectable, it's variable, and it can be reduced.
Take the manager who sees his female subordinates as eye candy and potential sexual conquests, and manages his department accordingly. According to the theory of patriarchy, this guy is a product of the society that created him, and his attitude isn't particularly surprising. According to the sexism interpretation, this guy is a dickhole.
More importantly, under the patriarchy interpretation, there's not much that can be done about the guy. He's just behaving like every guy under the system, more or less, maybe he's just a little more open about it. You could stop working for him I guess, but it wouldn't do you much good, you'd still be living under patriarchy either way. Whereas if you see him as sexist, you can go to HR and ask them to smack his ass down, and they very likely will. Or if they don't, you can stop working for him and work for someone who really is more respectful.
The problem with "patriarchy is like air," is, well, what the fuck can you do about air? You can't stop the air. Whereas sexism--sexism is like bacteria. There's always a certain baseline level everywhere, but some things are just infected, and here and now, with the tools we already have, we can Lysol those fuckers.