The difference between capitalism and socialism and mixed economies are pretty straight forward...

Capitalistic societies may have social safety nets to help those in need, but it does not have a stated goal of achieving economic "equity". Instead, the goal is largely to allow the economy to operate as a marketplace where owners reap the rewards of satisfying consumer demand and workers are compensated based on the value of the services in the marketplace.

Socialism at some level is about the government determining how goods and services are distributed and it generally has equity of outcome as one of its goals without respect to what each person contributed. Pure socialistic economies almost always fail due to the lack of incentive to produce, while the desire to consumer remains undiminished.

Mixed economies attempt to preserve elements of capitalism in order to encourage citizens to produce more and elements of socialism as they often go beyond a simple social safety net with regard to the government role in the production and distribution of goods and services. As an example, corporate welfare tends to be prominent in mixed economies as the government attempts to manage the production of goods and services besides simply helping those in need.