If someone claims money, they ask for it because it legally belongs to them.
1. People have to show their passports to get medical treatment, to claim welfare benefits, even to send their children to school.
2. You should claim your pension 3 or 4 months before you reach pension age.
If someone demands money, they ask for it in a forceful and sometimes aggressive way, even if they do not deserve it or have a right to it.
3. My mother used to watch at a window in case a landlord turned up and demanded money we hadn’t got.
4. They are demanding still higher wages.