Cure is used for illnesses and diseases, so you can be cured from measles. Once there is a cure for cancer, people will be able to be cured from it. For now, they can only recover, not even fully, as it can come back. Another way to get cured is to visit one of those miracle healers in church who instantly cure people from blindness and get them out of their wheelchairs. In that case they get cured very fast. Healing is a process, it takes time. Typically, you heal wounds, cuts, scratches and broken bones. You also heal emotionally, like from a loss of a loved one, you go through a healing process. To recover is to feel better physically so you can function normally. I guess you recover from a cold, you get over the cold, it's not cured or healed. If you have a car accident, first you would heal (your physical visible injuries) and then you would recover, meaning you can walk normally, your neck would stop hurting etc.

You recover from an illness like a cold. You may have some lingering symptoms like a cough, but you are able to work again. You heal a wound. If you cut yourself, skin slowly grows over the wound, it may be red for a short time, but when it finishing healing, you would not know that you had been cut. Many researchers are looking for a cure for cancer. If they find a cure, the cancer would be gone and the sufferers would be completely healthy.