Both are steroid hormones.
Cortisol is the more 'powerful' of the two, accounting for about 95% of the glucocorticoid activity in the body. (For a look at the extensive list of things that fall conveniently under the phrase 'glucocorticoid activity', go here.)
Cortisone is a metabolite of cortisone, but it has much less of an effect. In many regards, it may be perceived as an inactive form of cortisol, although cortisone can be converted into cortisol if needed.