I personally suffer from anxiety without depression, so it seems easy for me to tell the difference. An anxiety attack for me is something that comes on very suddenly, leaving me with an overwhelming sense of impending doom. Usually it's a feeling that I'm about to lose control or drop dead right then and there. The thought that often goes through my head is, "oh no, something isn't right, this can't be right, I have to call an ambulance" or "I have to get out of here, now!". When the attack passes, it is followed by feeling extremely jittery, weak and anxious for hours or even days. During the attack, any number of physiological symptoms present, including pounding heart, feeling like I might pass out, wind, diarrhea, feeling weak, etc.
I did suffer from mild and occasional depression when younger. It seemed to be more of a feeling of disappointment with various things in life leading to an extended malaise, but with a chemical origin (brain chemistry). (I think of such episodes as "chemical depression" as opposed to "ordinary depression".)
But because anxiety and depression can often go together, it is not usually so easy to distinguish them.
I read that sunshine and aerobic exercise help with both depression and anxiety (in my case exercise helped more than normal since I used to be overweight; a contributor to my depression at the time).
Treating either depression or anxiety seems to require a multi-pronged approach. Most important in my experience seems to be teaching yourself (or be taught) how to shut down unproductive thought patterns as soon as they occur. They are habits which apparently exacerbate both issues (not necessarily immediately, but cumulatively).
For anxiety, ruthlessly removing sources of anxiety (especially initially) can be helpful in the long run. Being blunt with people and simply telling them what the issues are that are stressing you out, and how much of an impact that is having on you, can help. Bottling up those issues and being unassertive and endlessly constructing "what-if" scenarios before or after the fact, exacerbates the problem.
For anxiety, cut out caffeine and other stimulants. Avoid intense anaerobic stress (e.g. heavy weightlifting or aerobic exercise past the point of exhaustion), which stresses the nervous system. Also keeping the system running smoothly (e.g. modifying your diet for smooth digestive transit and controlling any allergies with standard hayfever antihistamines) seems to substantially lower overall stress levels.
And obviously there are pharmaceutical options if you consult people qualified to dispense them.
At some point you begin to realise you haven't had any real attacks in a while and that you are recovering. That relief in itself helps in the recovery process for anxiety (which I have heard can take months or even years in some individuals).