I have a wisdom tooth that has broken and it is causing pain in my shoulder and neck... Is it infection or swelling? How severe is this?

Both. Infection causes swelling. Broken tooth infected. Infection is spreading through your body. From your symptoms I would say very severe. You need immediate dental treatment. This can't be cured with just a pill, or by waiting, or by ignoring it, or by a Physician. See Dentist immediately.