Is there a difference between a symposium paper and a conference paper
AFAIK, a symposium is essentially the same as a conference.  I think it tends to suggest a somewhat smaller size, historically, but definitely bigger and more competitive than a workshop.  For instance the ACM Symposium on Operating Systems Principles is one of the most competitive conferences in the computer systems field, and gets maybe 500-600 attendees these days.  Some "conferences" get many thousands, though some get only 50-100 too.  

Now, perhaps in other fields than computer science, there is a completely different definition, but if you have to pull the trigger, you should assume it is a conference and not a workshop.  

Besides, you submitted the paper and they accepted it.  It would be really poor form to pull out right now, even if somehow it didn't count toward your graduation.  Didn't your advisor know you were submitting it, and think it was a good idea?