New Right is an ideology that can best be summarized as a synthesis of neo-liberalism and neo-conservatism which are revived versions of their older counterparts.
Neo-liberalism especially supports this claim by having several concepts and notions that underline its connection with liberalism. Economic individualism is one of these notions, advocating an increase of individual freedom by eliminating government oppression, pushing individuals to be self-reliant; for example taxes should be kept minimal in accordance to this concept.
Another notion included in neo-liberalism is libertarianism, which was best summarized by Nozick’s beliefs, who pushed for extremely minimal government which would only be involved in the provision of security and life; if it is involved, that is. So libertarians see tax as coercion, forced labour, oppression.
But a key concept within the neo-liberal ideology is the revival of Austrian economics which consisted of Hayek’s beliefs that liberal-capitalist economy was made up out of a trial and error system, which was considerably better than market socialism in Hayek’s mind. This underlined the nature of New Right as a reaction towards Democratic Socialism.
The last important element of neo-liberalism consists of the ideas of Milton Friedman who believed that the only economic responsibility of the state should be the problem of inflation; in other words, monetarism. Monetarist concepts dictate that the government should only puruse a policy of sound money and sound regulations. So inflation would mean that the state is exceeding is limitations and assuming illegitimate roles which in turn means limiting the capacity of the state economically is necessary.
So neo-liberalism does not specifically differ from liberalism, it is actually a revival of it, including slightly different ideologies and concepts in this turn.