Harm is a physical or mental injury, and to cause that injury is to harm. Harmful describes someone or something that causes injury.
Ex: Before doctors begin to practice medicine, they take the Hippocratic Oath, which cautions doctors, above all, do no harm. Doctors take the oath seriously and do everything they can to prevent harming a patient.
Ex: Although you need some exposure to the sun to help your body use vitamin D, too much exposure is harmful to your health as shown by the dramatic increase in skin cancers.
To be in harm's way is to be susceptible to danger and usually describes those who have jobs like firemen, policemen, and soldiers. There's no harm in trying suggests that you might actually gain something from trying a new activity.
Ex: Firemen are local heroes and deserve our respect and admiration, as they put themselves in harm's way every time they answer an alarm.
Ex: Unless you have some serious allergies, there's no harm in trying new foods. You may discover your new favorite flavor to share with your friends and family.
Damage is harm done to something that reduces its value or usefulness. To damage is to do harm.
Ex: The damage to Kimberly's reputation will be hard to repair, especially because the person who accused her of wrongdoing had an editorial published in the local paper.
Ex: Each year, tornadoes damage or destroy entire towns in the Midwest. Despite the extensive damage, most residents clean up the mess and rebuild in the same place.
Damage control is the effort to reduce the effect of negative publicity, often regarding political campaigns. Collateral damage is an unintentional effect of caused by an intentional, destructive action.
Ex: When the press reported that our candidate had served time in jail as a young man, we had to do a lot of damage control in order to prevent his losing the election.
Ex: The type of coal mining known as mountaintop removal generates extensive collateral damage. Trees are knocked down, blocking streams that residents use for drinking and watering their crops.