These are quite closely-related experiences, aren't they? I think in terms of lions.
Watching a lion chase a gazelle is exciting. Being chased by a lion is fear. Having a similar feeling with no lion is anxiety. Something like that. The point is that context matters quite a bit in how we label these things.
Another point is how the term "anxiety" is often applied to normal variations in people to medicalize something. Example: "I am afraid to meet and mingle with people" vs. "I have social anxiety." Same thing, sounds different.
This is all terribly oversimplified. And so is this cute presentation below, but it makes some good points.