Well, I dunno abot organic eggs - Im not paying $5/dozen to perpetuate that hustle.
But I can speak about my own yard birds' eggs, which get only local mill feed, kitchen scraps and 'grown' treets. I'd call them 'natural', as opposed to "organic."
Flatly put, I haven't detected a drastic difference in taste. I 've tried, even doing side by side comparisons. There is a texture difference and "yard eggs" cook up more firmly. They dont spread out in the pan as much and the yolks are darker, and dont run the same. They seem to work better in batters too, giving it slightly more body. I know one woman who wont make cakes with store eggs.
And you cant hard boil them properly when they are dead fresh - they have to set for a couple of days.
But there isn't a notable taste difference, I'm sorry to say. I WANT there to be one and I try to come up with one, because I've put so much into them. But with my eyes closed, I cant tell enough difference in taste to make a difference.