For the time being, not much.
I mean, there's one like Jessica's Auntie that just makes you call her Aunt Jessica.
But most are just a milestone of some sorts, but from what I heard the idea is for the quirks to sort of block content.
Using Jessica again as an example, her "Adultress" quirk being on would mean that you'd get more chances of doing scenes similar to the one where you unlocked the quirk.