Burger here, I'm going to tech school for Sustainable Ag, and it's a big topic.
Organic Ag had become bastardized by the capitalist, a label to mark up prices that doesn't necessarily correlate to higher quality food or beneficial environmental practices. What it was bastardized from, which is now referred to as Sustainable Ag, is a mostly legitimate venture with both environmental and health benefits, and is very necessary in burgerland. Our food system is the most fucked up in the world.
Mostly, it has to do with the way the soil is treated, and the varieties of plants being used. Soil is a living organism, not a static material, and needs to be engaged with as such, otherwise you kill it, which is what happens on conventional farm. Also, in burgerland, we've fucked up the breeds of veggies we have. Since the 50s, we've bred them to be bigger, store better, grow faster. However, this is mostly water weight, and comes at the price of being really flavorless, shitty food. The distinction Heirloom is a result, because here we need to distinguish between old things that are good and new things that we flavorcucked.
However, to zizecs example of organic apples, he has a point. Since most fruit trees are clones, cut and grafted, they were exempt from the breeding degradation. Any flavor problems in fruit trees has to do with them being picked at the wrong time of year, or possibly soil quality issues. So organic apples won't likely taste any different (but won't be covered in pesticides proven to be carcinogenic).
Also, GMOs are a problem, but only because they enable porky to get away with fucking up the soil and spraying new pesticides on us, not because the are 'unnatural'