What are the legal definitions of damage and damages? Do they mean different things?
In the legal world, damage is defined as a loss or harm resulting from injury to a person, property or reputation. Damages, on the other hand, refers to compensation - such as a monetary judgment - provided to a person who has suffered a loss or harm due to the unlawful act or omission of another. The person at fault - the one who caused the loss or harm - must compensate (or pay) the injured party for his or her losses, i.e. he must pay his damages for the damage he caused.
Damages is an important concept under the law, because unless a party suffers damage (i.e. a loss), he will not be paid damages (monetary compensation). Picture, for example, a situation in which a contract is breached: Joe has a contract to buy a boat from Stan for $400. If Stan decides not to sell Joe the boat specified in the contract (aka breaches the contract) but instead gives Joe an even better boat for free, then Joe technically has suffered a breach of contract. However, because he did not suffer any financial loss or harm (aka he did not suffer any damage), he cannot sue and collect damages for breach of contract.
Proving damages is also important in a personal injury case. This can especially be an issue in medical malpractice cases based on missed symptoms of an illness. The plaintiff will need to prove that the doctor's negligence - i.e. missing the symptoms - caused him to suffer damage. If the doctor can successfully argue that the plaintiff's illness would have caused the exact same type and extent of damage even if it had been diagnosed properly, then the plaintiff will not be viewed as suffering any actual damage as a result of the doctor's negligence. Thus he won't be able to collect damages, even if the doctor was negligent.
For more information on damages and for assistance in determining what your potential damages may be after an injury, you should strongly consider speaking with a lawyer.