With panic attacks you feel terror, fearfulness, shortness of breath, heart palpitations and the anticipation of gloom and doom, or losing control of yourself.
These attacks often last only minutes, but they are so intense that afterwards, people become obsessed by the possibility that they will have another such attack…
… thus the "fear of fear" syndrome that makes panic attacks an anticipatory disease.
With social anxiety disorder, the anxiety is attached to social situations.
When you anticipate having to meet unfamiliar people, or when you expect that you might be subject to public scrutiny (even if that "public" is only one person), you feel similar symptoms to panic attacks.
With social anxiety disorder, the person learns that they can "control" their anxiety by simply avoiding social situations. They can end up severely limiting their ability to socialise and lead 'normal' lives in order to not feel the underlying anxiety.