An anxiety attack, people may feel fearful, apprehensive, may feel their heart racing or feel short of breath, but it's very short lived, and when the stressor goes away, so does the anxiety attack.
Panic attack on the other hand doesn't come in reaction to a stressor. It's unprovoked and unpredictable. And during a panic attack the individual is seized with terror, fear, or apprehension. They may feel that they're going to die, or lose control or have a heart attack. They have a host of physical symptoms which may include chest pain, shortness of breath, dizziness, nausea. And in addition to these terrifying panic attacks, people start worrying about having the next one. So there's a lot of what's called anticipatory anxiety.
Hope this helps!