But the notion that "fear" always connotes something bigger and stronger than "anxiety" breaks down in real life experience.
You can have a short-lived fear response to the bee buzzing around your face, and you can wake up at three in the morning awash in anxiety that won't let you get back to sleep.
When the distinction between "anxiety" and "fear" isn't critical to the discussion at hand, I use just one of these words as the umbrella term in my book, The Dance of Fear.
Anxiety, apprehension, fear, terror--however you name it, what matters is how you cope.
In everyday conversation, we use the language of emotions that we're comfortable with and that fits our psychological complexion. I've worked with clients who don't report feeling anxious or afraid. "I'm incredibly stressed out..." is their language of choice. "Stressed" is the codeword for "totally freaked out" for people who are allergic to identifying and sharing their own vulnerability.