You are mixing up 'neoliberalism' with libertarianism.

Libertarianism is the belief that there should be a separation between society and government such that government only protects the freedoms of the individual. Social functions of society should not be implemented by the government.

Neoliberalism is the belief in free-market capitalism as the solutions to fix all the ills of society.

The Republicans used to be the party of neoliberalism, and had a small segment which was libertarian. This is perhaps where your confusion stems from.

Now, the Democrats, since Clinton, have become neoliberal, and eliminated all the economic and social protections for the middle and working class. We essentially have two party system where both parties are neoliberal. The Democrats are a kinder, gentler form of neoliberal, but neoliberal none-the-less.

To give an example, look at Obamacare. It is a free-market solution to healthcare. It is neoliberal. Instead of providing universal healthcare by a government program, the Democrats provide a free-market solution.