Oh man, don't get me started. I have been eating organic food since I was pretty young, for a variety of reasons.
As a lover of food and an avid cook, the taste difference is unbelievable. As many have pointed out, you feel a lot better since your body doesn't have to work so hard to filter out all the pesticides and nitrates etc. Pesticides and unorganic fertilizers are some of the biggest polluters of the earth - the runoff into drinking water has long been known to cause cancer and leukemia and all kinds of problems. Usually you are supporting smaller farmers rather than large corporations. I don't agree that food you have to peel is OK, since the fertilizer that these foods are absorbing for nutrients are surely still present, but I haven't done much research.
As far as bananas go, I will never eat a non-organic one. Chiquita banana (formerly United Fruit Company) is one of the MOST notorious corporations in the world. Responsible for lots of exploitative neocolonialism in several Latin American colonies. I took a whole class in college about this subject, and it was astounding the power that corporations can have over whole countries (as we are seeing more and more in America today). And the taste difference in bananas is one of the most extreme.
As far as meat and dairy, I was vegan for many years mainly to protest the way that these animals are treated. I never liked the taste of milk growing up, but recently I started drinking some organic milk, mainly because I am allergic to a lot of foods and needed more calcium. I can actually handle the taste of it - it is SO much better than regular milk.
My last point would be, that so much of what we do in America is quite artificial - that is, we shop at Walmart for food that is imported fom around the world, we eat microwave meals and fast food with ingredients that have unpronouncable names...it's not "real" food. I like to think that the reason my great-grandma lived to be 100 years old is because she ate quite naturally. Organic food is more natural, because there are not all the additives that our body needs to deal with.
You are what you eat.
Sorry, I started and I guess I couldn't stop...