Since you are trying to discern a difference in terms of writing, no, there is not much of a difference. I would say that you can use these two words interchangeably in most cases.
Nevertheless, if we get really picky, we can pull the words apart and distinguish some difference between the two words. To feel is related more to a sense of physical sensation, for example, you can feel the warmth of the summer sun. So, if you think about it, it doesn't make much sense to say, "I feel happy." But people say that all the time. Happiness is an emotion, it is not a feeling. Emotions are connected more to a state of consciousness. And yet, it is awkward to say, "I am currently experiencing the emotion of happiness." We just say, "I feel happy."
Part of the reason that these two words are clouded together is that when we come to a point of intense emotion, like anger, we do "feel" something.
I guess that if you want to know how to use these two words, a metaphor might be helpful. Perhaps "emotion" is more technical and precise than "feeling" so we would use feelings in a poem, and emotion in a textbook.
What a great question! I feel that my emotions are really mixed up now.