1

They are both nouns and they mean

Indigents: Qui manque des choses les plus nécessaires ; nécessiteux 1

Démunis: Personne qui n'a pas de ressources suffisantes sur le plan économique et social.2

They are not synonyms so what is the difference between them?

2

Well to begin with, "indigent" (plural indigents) is used most of the time as an adjective and "Les indigents" is indeed the noun derivating from this adjective. Then, "démuni" also is an adjective, and I cannot think of something else but a mistake from Larousse because I've never heard of "démuni" used as a noun.

Now, the difference between those two ajectives, I'd say "démuni" designates people who barely have enough properties to survive, and "indigent" is used for people who are totally running out of money, they are the poorest ones.

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2

Indeed, indigent implies extreme poverty.

Social services can issue what is called a "certificat d'indigence" (certificate of extreme poverty). Its issuance is required for instance when a deceased person has neither relatives nor money to pay for its funerals. The city will fund the burial of persons holding such a certificate.

In jails, "indigents" are the people with no support from the outside. They have to do their own laundry in the shower room (usually detainees give their laundry to their visiting family).

"Démuni" (which I confirm is almost always used as an adjective) basicaly means that you are lacking something in a great way. 2 uses come to my mind : "Se retrouver démuni face à" : to face [something] powerless / helpless

Politics will often refer to "les plus démunis d'entre nous" : those among us who have the least [literaly the most resourceless among us]

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