What is the physiological difference between cortisone and cortisol?
Cortisol is directly given to the patient - it is then called Hydrocortisone. I don't know why this is done but probably to avoid confusion because both names are too similar.

The difference between both forms is one hydrogen atom at the C11 position of the molecule in Cortisol (or Hydrocortison) (image from here):



Otherwise is Cortisol (or Hydrocortisone) the active version, while Cortisone isn't very active in the human body.
Both forms can enzymatically turned into the other form (from active to inactive and vice versa). Since Cortisone needs to be activated in the liver, it can only be used for oral uptake. Most of the uses which are colloquially called Cortisone in fact contain Hydrocortisone.