What are the advantages and disadvantages of having a prime minister?
I assume that you are asking in this question about the pros and cons of having a parliamentary system, with a prime minister as the head of government, as opposed to having a system like that of the United States where the president is the head of government.
Perhaps the main benefit of a parliamentary system is that it ensures that the prime minister is more or less in control of the government. In a parliamentary system, the prime minister is chosen by the party that has a majority of the seats in the parliament. If no one party has a majority, the party that gets the most seats has the chance to try to create a coalition with other parties to make a majority. Either way, the prime minister is the head of the party that is in control of the parliament. This is good because it ensures that the prime minister will always be able to lead. This is in contrast to the US system where we currently have a Democratic president who cannot lead because Congress is controlled by the Republicans. In this way, we can say that it is better to have a parliamentary system because it guarantees the prime minister will be relatively capable of leading the country’s government.
Perhaps the biggest drawback of such a system is that it lacks checks and balances. In a parliamentary system, the executive branch is not independent of the legislative branch. Instead, the prime minister is elected by the legislature and answers to the legislature. This is in contrast to the US system where the legislature does not select the president and the president does not have to do what the legislature says. To many people, checks and balances are good because they make it harder for the government to be dominated by one party or one ideological group. The checks and balances make it harder to get the government to do anything, thus making it harder for the government to do things that would infringe on the rights of the people. This protection does not exist in a parliamentary system.
We can argue, then, that the parliamentary system is better because it is more efficient and worse because it lacks checks and balances that serve to limit what government can do.