Use 'damaging', i e. the gerund, when the subject or object can be replaced with 'doing that action'.
Damaging other people's property is really rude.
(Doing that action to other people's property is really rude.)
If you imagine the word 'damaging', you see a person performing an action - kicking something, hitting it with a hammer, etc
If you imagine the word 'damage', you see a an object - a break in a window, a scratch on a car, etc.