Damage has several meanings that include the idea of changing something for the worse, changing the integrity or value of something, doing something legally wrong to someone, or the cost of something.
My car was damaged in the wreck. (changing something for the worse)
She damaged her knee in the game.(changing the integrity of something)
Her lies damaged his reputation. (to affect negatively)
The damage comes to $498. (This is an idiomatic use and not usually used in formal writing).
Harm shares two senses with damage but has one that differs which is physical injury.
She harmed her knee in the game. (changing the integrity of something)
Her lies harmed his reputation. (to affect negatively)
Injure includes two senses that are similar to damage and harm, that is to affect negatively but includes one meaning it shares with harm.
She injured her knee in the game. (changing the integrity of something)
Her lies injured his reputation. (to affect negatively).
Shares only with harm
The broken glass injured his foot. (wound)
The senses come from WordNet.com
From these examples, it looks like damage has several meanings which makes it a good word to use with something that is negative and causes some type of change to something. However, when we write about a physical wound or a physical injury to a person’s body, then harm, injure, or hurt work better.
Damage does not refer to the wound, but it does describe something that is more long term and usually means an injury or hurt that results in a change in the body part or parts.