The evidence for this is invariably anecdotal - some organic food tastes better, people say, though that has nothing at all to do with structure or nutrition. I have no complaints with people who don't know any better falling for this stuff - 'organic' food is a multi-billion industry, so lucrative that the farmers most likely to be organic can't afford the fees and paperwork to get the little sticker that says they are organic - that's capitalism. But I like to puncture myths, especially if they are going after less informed people who just want a better life. Therefore, organic food, homeopathy and evolutionary psychology are recurring targets.
Some produce you buy from a farmer claiming to be organic may well taste better - it just has nothing at all to do with whether or not he uses deadly-but-natural strychnine as a toxic pesticide or a synthetic kind, and anything with an 'organic' label is allowed dozens of 'inorganic' ingredients, like gelatin, coloring and insulin. See the full list of synthetic/inorganic products allowed in both growing organic food and in the organic food itself.
In reality, what you think is organic is likely only 95% organic. No matter what the sticker says, if you do not personally know the farmer and visited his farm, you might be fooled. The reason there is a 5% exemption for 'essential' synthetic additives is because your eggs would cost $20 a dozen if a farmer tried to actually be 'organic'. Living on a farm in rural Pennsylvania surrounded by Amish people, even my getting milk with a ladle right out of the vat at the next door neighbor's was not 'organic' but it's as close to natural as you can get(1).
A more recent marketing gimmick has been that 'free range' animals are superior. Obviously there can be something to that. If you go to the mountains of Montana and shoot an elk, for example, the meat might taste a little 'sweeter' to you because elk eat fennel. So all you meat eaters who believe flesh tastes different based on how it is raised, you are correct. Those folks in Kobe know what they are doing.
But if you put a cow in the mountains is the milk going to be nutritionally different? It makes no sense to anyone who knows a ruminant digestive system. Yet some people believe free-range cow milk will be nutritionally different. And then we get to eggs and whether or not an egg produced by a chicken that walks around will be better.
A recent study conducted over a two-year period found 'free-range' eggs were not nutritionally different from a chicken kept in a cage(2). They used 500 pullets/hens and moved the ones for the range environment 12 weeks after hatching. After maturing, they collected egg samples at 50, 62, and 74 weeks and sent them to four different labs commonly used for egg nutrient analysis.
The results showed no difference in levels of vitamin A or vitamin E based on the hen being free-range or kept in a cage. Nutritionally negligible β-carotene levels were higher in the range eggs, contributing to the darker colored yolks observed in those eggs.
Citation: Kenneth E. Anderson, "Comparison of Fatty Acid, Cholesterol, and Vitamin A and E Composition in Eggs from Hens Housed in Conventional Cage and Range Production Facilities," Poultry Science, Poult Sci 2011. 90:1600-1608. doi:10.3382/ps.2010-01289
(1) Illegal, I assume, and was then, despite it being as natural as possible. That's the downside to big government. California spends billions subsidizing green energy yet a clothes line is illegal in virtually every housing plan in every city.
(2) As a bonus, the study found that eggs overall were not as bad as thought. Both cage- and range-produced eggs actually have lower cholesterol levels than previously believed, which has led the USDA to lower the cholesterol guidelines for eggs in the USDA Nutrient Database for shell eggs to 185 mg per egg, down from 213 mg.