We are doing something now, and I want to know if it has completed. "Are we finished?" We did something in the past, and I want to know it wsas completed. "Did we finish?" Я попробую писать по=русски. Сейчас мы делаем что-то, и я хочу знать если мы только что заканчиваем. "Are we finished?" Мы сделал что-то, и я хочу знать если мы уже закончил. "Did we finish?"

They can usually mean the same thing. For the most part, "are" is a present term, and "did" is a past term. So "Are we finished?" , can refer to at the moment asking someone if you are finished. "Did we finish?" , can refer to something in the past. It can be a minute ago or days ago. Keep in mind so long as the time is in the past. While eating dinner at a wedding you look to your cousin as you notice the plates being taken away and asked, " are we finished? " Three days ago you completed a task given to you by your professor. You worked on it with a fellow classmate. You seem to have forgotten if you finished it so you ask them, " did we finish the assignment? "