Country A is a federation, and Country B is a unitary government.
In a federation, power is shared between the central government and local/state governments. The central government has authority over certain issues, while local/state governments have authority over others. However, the central government does not have complete authority over everything. This system is seen in Country A.
In a unitary government, all power and decision-making authority are held by the central government, and local/state governments follow the decisions made by the central government. This system is seen in Country B.