What exactly is the point of the Add Another Answer button? What would we need to do with it that we can't do by adding another clause to our already existing answer?
If you have a really different approach to solve something, you can given another answer.
Why? Because our engine is designed to let the best answers float to the top. If you have two answers in one, it's unclear for others seeing the question which of the two solutions is the best.
So if you have a better solution than you previously had: answer it again
I believe it is there to remind answerers that they already have answered that question. On questions with lots of answers, or questions with lots of answers that have been revived after a multi-month fallow period, it's not impossible to forget you've already answered a question. It happened to me once :}