What conflicts might arise between capitalism and socialism?

Brief summary:

Capitalist society would believe that national institutions could be run by independent businesses, franchises and trusts, and the people pay them directly for it's use. The company has autonomy on how it's run. There would be a free market. Rewards often reaped by the successful, however benefit felt by many. Less focus on those who can't, than those who can.

Socialist society would believe that national institutions could be run by the state, with the people paying for the institutions through taxation, into the state. The state has autonomy on how it's run. There would be an open market. Rewards often reaped by the average person, however benefit felt by those in authority or power. More help for those who can't.