The vice-president is only a collaborator with the U.S President. The prime minister is more than that. France is a parliamentary system and the tradition is the president chooses the prime minister in the major party in the national assembly. If they are from the same party, it's difficult to separate their respective power because they are both working.
The separation of their power is clearly in "cohabitation" when the prime minister not come from the presidential party. France had three governments in this case during the five republics.
In this situation, the prime minister does all, but diplomacy, military affair and public order stay in the president's hands, and he can choose the ministries from those administrations, but the prime minister leads all the other ministries, even the very influential ministry of finances.