I hope you know the difference between a President and a Prime Minister.
Vice President is someone who can play the role of a President when he is momentarily absent for some reason. Vice president is mostly a constitutional position. For instance, in the USA, a president has to submit his resignation to the Vice President, a vice president is in charge when the President is on a vacation. In India, a Vice President is the ex officio chairman of the upper house of the parliament.
Deputy Prime Minister is someone who can play the role of a Prime Minister when he is momentarily absent for some reason. Deputy prime minister is mostly a non-constitutional position who is generally a holder of an important portfolio like Ministry of Home Affairs, etc. For instance, in India, there hasn't been any Deputy Prime Minister since 2004.