Claim and demand are both legal terms requesting certain things, however the two is not interchangeable, as demand refers to a summons to court, while claim is used to demand compensation for something.
In everyday use the difference between the two verbs is that demand expresses a forceful request, while claim is to demand ownership of something.
Demand may also refer to the desire to purchase goods.
Claim: A monetary amount to be paid by the requestor in a civil action based on the assertion of a right.
Demand: Requirement that something be performed or made based on a request by the other party.
