It really depends on the birth control itself.
I'm on the generic for my shot, and I haven't noticed any difference when I switched from name to generic.
Now, working in the medical field a LOT of patients noticed a big difference when they switched from Yaz to the generic, they had awful headaches and high blood pressure and some even couldn't eat or would throw up if they did. Others tolerated the generic better than the name!
So I think it depends on the person as to what you tolerate and what you don't.