A president can only exist in a Republic, where they are the head of state and independently appointed from the government. A Prime Minister is the head of the government (ie chosen from within the majority party by his/her colleagues) within a monarchy.
Thus George Bush is (theoretically!) elected directly by the people. On voting slips for the General election his name will appear. However, in the UK, when you vote in an election, you vote for your constituent MP, which - unless you live in County Durham - will not be Tony Blair.
An old way of describing the role of the Prime Minister is 'Primus Inter Pares', which means First Among Equals.