A state is a political corporation, an association of people or other units who have organized together for mutual benefit.
A government is the entity responsible for ruling a state (or a political sub-division thereof).
In other words, a state is like a club, while a government is the officers of said club. Governments come and go, but the state remains.
While state and government are separate things, a state cannot exist, usually, without a government, and a government cannot exist without a state. One example of this point in practice comes in the declarative theory of statehood. According to it, a state is only sovereign if it is in possession of four qualities: (1) a permanent population, (2) a defined territory, (3) the ability to enter into relations with other states, and (4) a government.