The Prime Minister is the chief of the government and the leader of the majority of the parliament, while the President - like a Queen - is the head of state and represents and annoints people like the court of justice. Two reasons for this: first nobody can do these two jobs at once, because already one job is very straining and stressful. Furthermore in a democracy powers have to be divided to avoid abuse of power and accumulation of power (e.g. Napoleonic France, Nazi Germany). Additionally a head of state must be neutral, a thing the prime minister as a leader of a party simply cannot be, because he is too involved in party politics.
Mind you: systems differ: in France the President (who was elected by the people) has the power and selects the prime minister (chief of government) and can fire him/her whenever he likes. In Germany ("Kanzler" = chancellor) and Italy the prime minister has the most power. He/She is elected by the voters. The President is only the head of state and elected by the parties. In USA the president is again elected by the people and is also head of the government (the Americans tried in the first decade without a president, but it failed) and in the UK and other Commonwealth the Queen, who is not elected, is just a neutral head of state, while the Prime Minister as the elected chief of government is answerable to the voters.
In Britain, the leader of the biggest party in the parliament is prime minster, never mind how many percentage he gets.
The difference between the UK and USA is just that, the US president is head of state AND head of government (dependent and answerable on voters), while in the UK the prime minister is only head of government (voter-dependent), and the Queen as head of state is NOT answerable to the voters.
-British PM's cannot hire and fire the cabinet as the US president can (because the PM needs his own party behind him. If his own party turns against him, he loses the job. US Presidents only loose the job, when voters elect somebody else)
-PM's can be vetoed by their own government, presidents cannot
-US presidents are formally commander in chief,PM's are not
-the strength of the PMs depend on the individual character of the PM, the strength of the US president lies in his constitutional role
Putin has more power, because he has accumulated more power when he was president and he just swapped his role for that as a prime minister, because he was not allowed by legislation to be in such a powerful position for a longer period. Therefore he just gave up his president role to his friend and supporter and by this he ensured that he won't get fired as prime minister. Clever, isnt it ? (and dangerous)