There are different ways of looking at organic foods, but here’s what I think.
If you’re talking about locally-grown foods, let’s use a carrot as an example. Since vegetables derive their nutrients from the soil, you would have the choice of an "organic" carrot grown in living soil, full of natural microbes and minerals, vs. one grown in dirt laced with chemical fertilizers. To me, it’s only common sense that the organic carrot would be better for your body.
As for "certified organic", that doesn’t mean much as far as I’m concerned. I know several local farmers I buy from that won’t become certified because they would then have regulators breathing down their necks, which according to them, would actually result in an inferior product. These people take great pride in the produce and animals they raise, and it’s as natural as anyone could wish for.
When you look at commercially produced "organic" vs. regular products, perhaps the difference is not as great. The benefit I see is that with the organic label, they can’t fill the product with chemicals that can’t even be pronounced. For example, I regularly buy Heinz Organic Ketchup, not because I think the ketchup itself is healthier being organic, but because it is made with real sugar instead of HFCS. Much better flavor, and no strange chemicals. Likewise with Swanson Organic broths.