What is my problem? The left side of my mouth goes to one side while smiling and while talking in public.

This is difficult to answer without photos. Everyone has some asymmetries. I recommend getting a formal consultation. Fillers can be used to improve lip symmetry. I hope this helps. 

Thank you for your question.  Sorry to hear about your concerns.  It is difficult to comment on your case without seeing photographs.  Based on your narrative, I would recommend that you see an expert for an in-person consultation and examination in order to make the correct diagnosis and treatment.  Hope this helps and best wishes!