This guide explains the method that a Fortune does on an axe. It increases the chances for you to get much better things like a sapling stick and many others including fruits as well.
If you put fortune on an axe then it boosts up the likelihood, you can get sapling sticks and even apples out of the sticks.
Breaking it with hands only yields regular drop but on the other hand, if you use fortune 3 on your axe then there is a probability of dropping saplings and sticks. The chances of getting apples also increases if it is an oak tree.
Fortune also affects mellons and seeds of the beet roots. Carrot, Potatoes and seeds of wheat and grass as well. It doesn’t mean that you are going to get these things at every chance. It just increases your chance but depends on your luck as well. You can get tons of mellons from it.