There's something of a tradition of joking about the guest of honour in Britain; it's to be found at its simplest level in the best man's speech at weddings, and flourishes in after-dinner speeches and at show business dinners.
The President of the United States, 'POTUS', holds a different position to our Prime Minister. Americans have 'respect for the office of President', whoever holds it. He may be criticised for his decisions on policy and how he acts in government but there is certain reserve about other comment. The Correspondents' Dinner provides an opportunity for him to be self-deprecating, to make jokes about colleagues and employees (which he would not ordinarily do) and for the press and others to make jokes about himself. There are certain conventions; GWB made a joke about looking for WMD, hunting around his office, which was far too close to politics, and sensitive as a subject and, though intended as a joke against himself, therefore failed and was controversial