To settle an argument, please tell me this. Are organic bananas better for you than ordinary ones?

No. Regrettably, so far there is no proof whatever "organic" foods are any better for one's health. Also, please bear in mind there is no legal definition of the word organic. Any vendor can cheerfully stick a notice saying "organic" on a product. Accreditation by the Soil Association is what you need to look out for if you do choose organic.