These two services work closely together within the same Office, because academic, and personal issues are often so closely intertwined. In general, you should make an appointment with the Academic Support division if your concerns are primarily academic (grades, writing papers, time management, exam skills, hiring a tutor, preparing for comprehensive exams, etc.).
You should make an appointment with the Counseling Office if your concerns are primarily personal (relationship difficulties, depression or anxiety, coping with a physical, learning or psychological disability, eating or body image, difficult life events or issues, sexual orientation, coping with a loss or trauma, culture adjustment, special needs accommodations, etc.). Both divisions are aware that students often have related concerns in more than one area, and hence collaborate together in order to identify the issues affecting your academic performance and devise a plan that will allow you successful go about your journey.