Insist means to demand something, as in "I insist that you finish this work by tomorrow." Persist means to continue something, as in "If you persist in slacking off at work, you won't finish the work by tomorrow."

If a person INSISTS that you answer a question, they would demand it in a FORCEFUL way. They might say, "I want an answer right now. I am not leaving until I get an answer". If someone were to PERSIST in asking you a question, they would ask you REPEATEDLY. For example, they might say, "Why did do you do that? Can you explain it to me? What were you thinking? What reason you could have for doing that? Can you explain the motive for what you did? Hope this helps

“persist” just means to last. It isn’t restricted to a person asking for information. “If the horrible weather persists, we’re going to have trouble meeting our production quotas” “If the political unrest persists, investment will surely drop”

If I can add, Sometimes it's not always demanding. If someone says "you can go ahead of me when waiting for something" Sometimes people can say, "oh no please I insist" https://www.italki.com/question/47655 So it can be softer than demanding

To persist means that you don't give up. To insist means you're forcefully telling someone you want them to do something