Schicksal is what is happening to you and you don't have any influence on it; it is most often used for something negative (like a terminal illness). It also has always already happened to you when you notice it.
Bestimmung means someone is convinced that you are meant to do/to be something (like become the next king), and that someone can very well be you. So you can actually choose to have one, like deciding it is your Bestimmung to live in a convent. You usually have to follow it (or you want to follow it if it's your idea in the first place), but unlike Schicksal, you can also try to avoid it.