They have very similar meanings. In my mind, prone is more incidental and possibly more negative. Someone prone to doing something might not be aware they do it often ("He is prone to running late") or it might be something they don't have control over it ("I'm prone to getting acne"). Inclined is more intentional. Usually when used with people it actual means they want to do whatever it is they are inclined to ("I am inclined to go to the party"), but it has a pretty formal sound to it. The words are more similar when talking about objects instead of people, and basically 'to be likely to' or to 'to tend to', but I would still say prone has a negative sound ("glass is prone to breaking"), and inclined has a formal sound ("That is inclined to happen"). I hope this was helpful!