There are two differences that distinguish the use of these verbs. 1) The original verbs have different meanings. "quedar" means "to be left", "permanecer" means "to stay". 2) The use of the "quedarse" is far more common than that of "permanecer", though, they convey the same idea, the latter is perceived as a somewhat formal. I would send a letter to my brother and say: Me voy a quedar en Italia dos semanas más. And I would send a letter to my boss and say: Voy a permanecer en Italia dos semanas más Both meaning: I will stay in Italy for 2 more weeks. If I wrote to my brother: Voy a permanecer en Italia dos semanas más. My brother would assume I am stressed and I am speaking in a "office" manner. If I wrote to my boss: Me voy a quedar en Italia dos semanas más It would imply that I have certain closeness, not friendship, with my boss, but I would avoid using this with a client if I want to stick to a business protocol. I hope this helps.