"Besides" is an exclusive term. In the sentence, "Besides chocolate, what flavor of ice cream do you like best?".. the asker wants to know about everything EXCEPT chocolate.
"In addition" is when something else is added to the list or discussion. As in, "In addition to your regular chores, I'd like you to clean the bathroom." This means you will do your regular chores AND clean the bathroom as well.
Sometimes people use "besides" when they should use "in addition", as in the last example, "Besides your regular chores, please clean the bathroom." But this is not the proper use of this word.