This is kind of two questions in one.
1) What is the difference between natural and organic?
Natural: Existing in or caused by nature; not made or caused by humankind.
Organic: Of, relating to, or derived from living matter: "organic soils".
From these, we can see they are quite different. A hurricane, for example, is natural as it is not caused by man. The same is true of rocks, suns, and the molten core of the earth. None of these things are organic as they are not living matter, made from a carbon base.
Can something be organic, but not natural? Sure. Imagine a cloned sheep—this is organic, but not natural. Or at least, one could argue that.
And the second question...
2) Why organic food and not natural food?
You are absolutely right that the phrase natural food would work pretty well for what we call organic foods today. That is, they are natural because they are not exposed to non-natural/artificial/synthetic chemicals. Organic doesn't fit very well because vegetables are organic whether or not they have pesticides applied to them.
As for the etymology of the term, the Organic Food Blog proposes:
"Where did we get the term “organic” food? Apparently in the early 20th century Walter James the 4th Baron Northbourne wrote a book called Look to the Land and coined the term organic farming as a holistic, ecologically based approach to farming in contrast to what he called chemical farming. Walter was a multi talented man being an agriculturalist, author and ever competed in the 1920 Summer Olympics where he won the silver medal."
In my view, this is a classic case of a phrase just being made up, even if it doesn't make sense, to avoid confusion with casual speech. That is to say, one might casually mention natural food, but organic is a rare enough word that it is unlikely that one would ever say organic food unless they were specifically referring to the phrase. Often, new phrases are introduced just so they don't sound like pre-existing phrases, if that makes any sense. I also have little doubt that some marketing effort went into determining that this was a favorable phrase.