Oof, that’s a tough question to answer in here. There is no really good way to generalise who has what power, and there is probably many ways to split the powers in a meaningful way.

You can read the articles on both positions specifically for France, which I do think in this case is a great example, on wikipedia, although if you want a more precise and complete understanding you’d probably have to read the french article and translate it.

The main advantage of this system is that when the president doesn’t have the majority to support him in the parliament, most of the executive power de facto shifts to the prime minister, who is usually nominated (by the president) in accordance with the parliament’s majority coalition. When that’s not done, the parliament can move to “censor” the government and force the president to nominate a new prime minister, who then nominates the rest of the government.

That system is a good way to make sure the president doesn’t do whatever the fuck they want if the parliament disagrees.