If I ever choose to eat organic (which to be honest, isn't often due to cost), it's not because I see it as more nutritious, it's because it's been grown without pesticides, fertilizers or better (read: nutrient rich) soil.
There's no surprise, the study was conducted by the American Journal of Clinical Nutrition, so it's going to be one-sided stating it's not necessarily a nutrient-superior food. A similar study, done by the U.S-based Organic Centre* found certain levels to be higher (antioxidants - very important! like vitamin C and vitamin E). Since it's grown in better soil, the food, in turn, would automatically have better nurtrients (nitrogen, phosphorus).
For my Canadian counterparts, there's also the Canadian Organic Growers, and good news! Organic legislation has been put into effect (finally!) in June! (There was no regulation before and it only took 18 years..heh.) So now anything labelled 'organic' has to adhere to certain standards. Familiarize yourself with the labels here and read more about it here, containing FAQ's.
Geez, I got a bit off topic, didn't I?
*the Organic Centre actually disects the American Journal of Clinical Nutrition's study and does comparative tests and their own studies. You can find the article here. It's their FSA response and very interesting!