I've had both separately and at the same time. Anxiety causes me to have a panic attack, but its the depression that causes me to dwell and feel trapped by the anxiety. Otherwise with no depression I can easily shrug it off and continue as normal.
If I could put it into words, I'd say I felt more scared than depressed. "What if this is how my life will be from now on? I don't see how I could possibly cope. This must be what torture feels like."
(Fortunately the reality is that I managed to stabilise my anxiety using some of the methods I mentioned, to the point that I have few if any discernible symptoms for periods of months at a time.)
I've actually read papers by psychologists who seem genuinely unaware that quite severe anxiety disorder can present without any significant depression.
However, when I was a teenager I had a form of OCD that psychologists at the time considered to be a holy grail, essentially untreatable. I taught myself some mental tricks, which I later learned are standard fare for psychologists (CBT basically, although I'd never heard that term), and it went away after two years. I really and honestly thought full recovery wasn't possible at the time on the basis of what psychologists thought.
With anxiety, I choose not to own the condition. I don't feel attached to it in any way. I view it as definitely temporary, even if it does still flare up occasionally. And I believe that full recovery is realistic based on my past experience.
Of course I am just lucky that I don't have clinical depression to go with my anxiety. But I have friends who have fully recovered from that.
That's incredible to me. Since this is mostly my condition.