To 'damage' is to partially cause harm or injury to a person or property, whereas to 'destroy' and to 'ruin' is to totally and completely spoil, demolish or devastate someone or something. Example: The looters damaged the shopfront with bricks and metal poles. The opposing army was totally destroyed in the battle. After the hurricane, the town was in ruins.

Damage and ruin can also be nouns. Destroy is only a verb.