A mixed economic system is a system that combines aspects of both capitalism and socialism. A mixed economic system protects private property and allows a level of economic freedom in the use of capital, but also allows for governments to interfere in economic activities in order to achieve social aims.
They were different, because in Thinker's case it was ruled in the favor of the students. They had the freedom to publish whatever they wanted, as long as it was not disruptive . In Hazelwood's case it was ruled in favor of the school, because the educators had a power to decide what could be published in student's newspaper.