The major difference is that a socialist system is more of a command economy while a free market economy is a market economy.
In a command economy, the government decides what things will be made, how they will be made, and who will get them. This is often done through such things as five year plans that set out goals for how much of which particular products will be made.
By contrast, in a market economy, it is the market that decides these questions. The government does not decide how many computers will be made or how many cars. It does not have much input into how they are made or who will get them. Instead, the market decides this through the processes of supply and demand.
So, the difference is that in a socialist economy, the government makes most of the decisions about the economy while in a market economy, the market (which means consumers) makes those decisions.