Anxiety and fear are two different levels of the same emotion. Anxiety tends to be irrational concerns, unease, or nervousness. Whereas fear is basically the next level up.
Personally I have always classified anxiety disorder and panic disorder as "irrational fear" because that's the easiest way to explain it to someone that doesn't suffer from either. But if you really want to dig down deep, there is a difference, imho. I can only speak directly to panic disorder, and my experience with it is best described as a constant "fight or flight" response. I'm not sure that's properly defined as a fear, but rather an instinct devoid of an actual emotion and a distinct feeling unto itself. I've often thought that (at least in my case) an instinct is being mistakenly triggered by a chemical imbalance and the reason you can't break free from it is that you can't resolve the fight or flight question. This ultimately leads you into a downwards spiral of repetition as the fight or flight instinct (correctly) triggers increased heartbeat, Adrenalin production, narrowing vision, and any of the other symptoms of anxiety / panic. These other symptoms, in turn, create another instinctual response, creating worse or more symptoms. This carries on until your body finally reaches a point of fatigue in which the fight or flight response can no longer be triggered.