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.