My perception of the difference is that emotions are our physiological responses to stimuli - changes in muscle tension, heart rate, stress hormones etc... and that our feelings are our perceptions and interpretations of those physiological responses. We are able to consider and reflect upon our feelings, and consciously choose whether to act upon them or not.

I think animals experience emotions much like we do in that they have a physiological fear and aggression response, but they are unable to make the step to consider or reflect upon those physiological changes, and only act upon them instinctively.