The verb treat is used for giving medical aid; making efforts to help a sick or injured person get better:
The doctor treated the girl with antibiotics.
Heal is typically used for injuries and wounds (not diseases), when the injury gets better naturally by the body's own processes:
It was just a minor cut, it healed in a day or two.
People wear casts to keep the broken bones in the right position while they heal.
Cure is typically used for diseases - meaning when medicine completely eliminates the disease and restores the person to health. It can be both a noun and a verb:
Antibiotics can cure minor infections.
The doctors said her cancer was cured after many weeks of chemotherapy.
Scientists have not yet found a cure for AIDS.
The verb recover describes the process of a person resting and getting back to full health after a disease, injury, or surgery:
It took him a month to recover from the injuries he suffered in the car crash.
I have a really bad cold and I'm staying home from work today to recover.
She spent a week in the hospital recovering from her kidney transplant.