Cortisol binds to mineralocorticoid receptors in addition to glucocorticoid receptors, thereby promoting renal retention of sodium and renal loss of potassium. It is ordinarily converted by the kidney to cortisone, limiting its effects on the renal tubules, but in cortisol excess, mineralocorticoid activity is observed.
I think this partial Wikipedia entry also succinctly summarizes the relationship/differences between Cortisol and Cortisone:
Cortisone is one of several end-products of a process called steroidogenesis. This process starts with the synthesis of cholesterol, which then proceeds through a series of modifications in the adrenal gland (suprarenal) to become any one of many steroid hormones. One end-product of this pathway is cortisol. For cortisol to be released from the adrenal gland, a cascade of signaling occurs. Corticotropin-releasing hormone released from the hypothalamus stimulates corticotrophs in the anterior pituitary to release ACTH, which relays the signal to the adrenal cortex. Here, the zona fasciculata and zona reticularis, in response to ACTH, secrete glucocorticoids, in particular cortisol. In the peripheral tissues, cortisol is converted to cortisone by the enzyme 11-beta-steroid dehydrogenase.
Cortisone has marginally reduced glucocorticoid activity compared to cortisol (80-90%), and thus, cortisone can be considered an active metabolite of cortisol. However, 11-beta-steroid dehydrogenase can catalyze the reverse reaction as well, and, thus, cortisone is also a precursor molecule of cortisol. Cortisone is activated through hydrogenation of the 11-keto-group, and cortisol is, thus, sometimes referred to as hydrocortisone.
Please let me know if you have any other questions. I hope this helped.