Wikipedia mentions this regarding deputy prime minister:
The position is often likened to that of a vice president, but is significantly different even though both positions are "number two" offices.
However, it doesn't explain what are the differences.
Can someone please explain the differences?
This may seem to mostly restate the difference between a prime minister and a president. That's because a large part of the difference between a deputy prime minister and a president is based on that difference.
In a parliamentary government, voters choose representatives. Those representatives vote to create the government and choose ministers. The deputy prime minister is chosen by parliament (or by the prime minister, who is always chosen by parliament), not the voters directly. As a result, the deputy prime minister can generally be replaced at any time, like any other minister.
In a presidential government, the president and vice president are elected by the voters directly. The vice president can only leave office by death, resignation, or impeachment. If the vice president leaves, there may or may not be a replacement process.
For example, in the United States, for most of its history, a vice president could not be replaced except by a subsequent election. Modernly the constitution was amended to allow the president to nominate a new vice president in case of a vacancy. The nominee could then take office with the approval of a majority of each chamber of Congress.
If a president dies or otherwise leaves office, the vice president ascends to the office. If a prime minster dies or otherwise leaves office, parliament may choose a new prime minister. There's no requirement that that person be the deputy prime minister. Parliament can bypass the deputy and pick any other qualified person.
A vice president doesn't need to have any powers. They're just spare presidents. John Nance Garner once said that the US vice president had as much power as a bucket of warm spit. A deputy prime minister is chosen specifically so that the prime minister can delegate to the deputy. If the prime minister is not delegating any power, then there is not much purpose to having a deputy.
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.
As an example, both Michael Haseltine and John Prescott were First Secretary of State, and Nick Clegg was both leader of the Liberal Democrats and Minister for Constitutional and Political Reform.
So in the UK, the deputy prime minister will be an important political personality independent of being deputy prime minister, and will stand in for the prime minister when needed.
To clarify: it’s not that the deputy prime minister has another job, it’s more that some ministers or in one case a party leader is also given the responsibility to be deputy prime minister, having to stand in for the prime minister should it become necessary.
In case of death or incapacity of the leader, the deputy would only be in charge temporarily, until the MPs vote for a new prime minister.
