Although besides can have many definitions, it's usually defined as 'other than; except' (e.g. I don't have any family besides my parents. There's no one here besides Bill and I.) Basically when you're trying to add an exception to what you previously said. I don't have any food (statement) besides this granola bar (exception.) In addition usually means 'also, as well as' (e.g. members of the board were paid a small allowance in addition to their normal salary.They study their instruments and, in addition, theory and music history.) So you're adding an extra person, thing, or circumstance. Like here for example... In addition to the recipe ingredients, you'll need the following (here 'the following' is what's getting added.) Members of the board were paid a small allowance in addition to their normal salary (an allowance is getting added.) Of course, besides can have other, less common definitions. Sometimes in addition and besides can be interchangeable. English is actually quite confusing, but I hope this helps.

Besides is used when the second point you're making isn't closely related to the first. In this sense it's similar in meaning to 'anyway'. 'In addition' is generally used when the two points go together to form one arguement. In informal language there can be some overlap in the meanings of the two.