How do communism and socialism differ in terms of the role that government plays in the economy ?

In a way, communism is an extreme form of socialism.

Socialism is often used interchangeably with communism but the two philosophies have some differences. Most notably, while communism is a political system, socialism is primarily an economic system that can exist in various forms under a wide range of political systems.

Under communism, there is a Centralized government, planned economy, the dictatorship of the "proletariat", common ownership of the tools of production, and no private property. Equality between genders and all people, international focus.  

While under socialism it exists in calculation that is as following;

Collective ownership, Cooperative common ownership, Economic democracy Economic planning, Equal opportunity, Free association, Industrial democracy, Input-output model, Internationalism, Labour voucher, Material balancing.

huh?

Explanation: