What is the difference between hydrocortisone and cortisone?

Hydrocortisone and cortisone are both corticosteroids. However, cortisone requires conversion in the liver to hydrocortisone AND is not active topically.
The relatively moderate anti-inflammatory potency of hydrocortisone makes it a useful topical corticosteroid for management of inflammatory skin conditions, because side effects (both topical and systemic) are less marked.

In our body, adrenal cortex normally secretes hydrocortisone (cortisol) which has glucocorticoid (anti-inflammatory) activity and weak mineralocorticoid activity. It also secretes the mineralocorticoid, aldosterone, which regulates water and electrolytes in the body.

Corticosteroids with high glucocorticoid activity is in itself are of no benefit unless it has relatively low mineralocorticoid activity, so that the effect on water and electrolytes is not also increased.

Cortisone and hydrocortisone have a relatively high glucocorticoid activity, which would result in fluid retention, making them unsuitable for long term use.

Cortef is a trade name for the drug hydrocortisone.