Consider the definitions:
2 Undergo or suffer (something unpleasant, especially an injury)
Become subject to (something unwelcome or unpleasant) as a result of one’s own behaviour or actions
1 Cause (something unpleasant or painful) to be suffered by someone or something
If the damage was malicious (someone else purposely caused the damage), then you might the damage was inflicted on them. If they brought the problem on themselves, then you can talk about the damage incurred. If the damage was caused by a natural catastrophe, it was simply damage sustained by them.
The terms damage incurred and damage suffered may also be contrasted in a legal sense, as described in the article When Does "Incurred" Mean "Incurred" for Insurance and Reinsurance Purposes? by Larry Schiffer. Note that the term damage may have a more narrow / specialist legal definition here than in the usage described in the paragraphs above.
And in 515 Ave. I Tenants Corp. v. Gutman Mgmt. Co., 29 Misc. 3d 1228A (N.Y. Sup. Ct. Kings Co. 2010) (citations omitted), a Brooklyn trial court distinguished between damages suffered and cost or damage incurred. The court stated that "suffered" means paid while "incurred" means become liable for.
To resolve the appeal, the circuit court had to construe the meaning of "incurred" under New York law. New York law, held the court, defines "incurred" for insurance purposes as "to become liable or subject to." Thus, found the court, liability for a charge begins at the time of the treatment for which the charge is imposed. The court held that the insured may be considered liable for a charge even if the insured does not ultimately pay that charge in full or in part.