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From what I learned about Neo-Classical and Neo-Liberal economical philosophies in different sources such as documentary series or Wiki site articles I understand both are "neo" versions of Classical and Liberal economy, and that all four are branches of capitalism,
but how would you explain the essence of difference (if there is one), between Neo-Classical and Neo-Liberal?
To expand on my comment. Your question is not really about economics but about philosophy or political science. The reason for this is that the Economics is usually defined following the definition of Lionel Robbins as: "Economics is the science which studies human behaviour as a relationship between ends and scarce means which have alternative uses."
Neo-liberalism is an ideology. It is a set of values or set of ends that based on the given values are worth pursuing. For example, from Neoliberal value of freedom one could derive the end goal of free speech. But economics does not examine either ends or values. Economics treats ends and values as given by an individual. Hence there is not really anything like Neo-liberal economics (although I seen the term used as a pejorative, like trickle down economics - but those are not really economic terms that have any meaning in economics - you generally wont find them in papers or textbooks or conferences) as that entails value judgments that can only be judged based on philosophy not science.
There is a Neoclassical school of economics. However, it has nothing to do with capitalism. Capitalism is a term that’s poorly defined and not used in economics. In most economics textbook you would not even find the word, outside some inserted exempts of newspaper articles in undergraduate books. Neoclassical economics is an umbrella term so its easier to classify theories. The term for example implies that models used will assume rational agents (as opposed to lets say behavioral school that relaxes this assumption). This being said the classification is used mostly by historians of economic thought as in present day its not uncommon for people to use different models and set of assumption to fit given purpose, so there are many economists who use neoclassical model in one work but behavioral model in another.
Also, Neoclassical economic models have no relationship to idea of capitalism (many neoclassical models suggest quite large role for government intervention in economy or redistribution given certain welfare function). For example, Jozef Stiglitz's work is Neoclassical, meaning almost all his papers use neoclassical models, but he does not like the idea of 'classic capitalism'. Another example would be Greg Mankiw, who mostly works with Neo-Keynesian models in his research, but personally he is supportive of the idea of 'classic capitalism' - his ideology could be described as quite libertarian or Neo-liberal.
