Cure is mainly used in medical world where the treatment is concentrated on the particular part of the body which is problematic through medication and other procedures hoping that it will get cure.
Healing is the holistic way of reaching to the deep root cause of the problem. Reiki or other healing modalities do not isolate the problematic part of the body but concentrate on the whole body and help restore balance and harmony to mind, body and soul bringing healing to the whole body.
Medicine and Reiki (other alternate healing modalities) go hand in hand. It will be wonderful to have a holistic medical approach.
It is not so common in medical world to have a holistic approach but there are many doctors who have adopted this approach as they are aware of the added benefits of Reiki and other healing modalities. One of the name worth mentioning is Dr. Mehmet Oz.