What is the difference between å existere and å finnes ?Feel free to just provide example sentences.

*eksistere

"finnes" is sort of softer and vaguer, like "be", whereas "eksistere" is equivalent to "exist", or is perhaps even stronger. "være" is often used in contrast to "finnes" to mark the same distinction as "it is/are" and "there is/are". E.g. "Det er/finnes ting selv ikke de forstår." = "Those/there are things not even they understand.".

"eksistere" is linked to matters of existentialism and such. E.g. Matematikere drøfter hvilke kategorier av tall som virkelig eksisterer, og hvilke som f.eks. kun brukes for å forenkle ligninger uten å ha noen betydningsfull valør i seg selv. = Mathematicians debate which categories of numbers truly exist, and which are only used to, say, simplify equations, without possessing any meaningful value in and of themselves.

"eksistere" is reasonably rare, as there are multiple ways to express similar ideas. E.g. there's the verb phrase "være til", which is used in cases like "Jeg elsker å hvile i sola og bare være til." = "I love to rest in the sun and only live/be/exist.".