Neo-marxism is basically just a more recent off-shoot of Marxism. It was developed after the first world war by the Frankfurt school and their main aim was to bring Marxism "up-to-date". They agreed with Marx in the sense that Capitalism is the root cause of the class-conflict and that all governments create laws and regulations that protect and promote the interests of the rich (the bourgeois as Marx called them). They also agreed with Marx and said that "the rich" maintain their power by promoting "ideologies". I would like to point out here that for Marx ideologies were BAD. He saw them as a means for the rich to control the poor or to make the poor remain happy in their current powerless situation without revolting since as you know there is power in numbers and the poor have always and will always out-number the rich. A neo-Marxist named Antonio Gramsci termed this "superiority of ideas as a means of control" as Hegemony.

The neo-Marxist just took Marxist ideas and applied them to culture and sociology rather than the purely economic setting they had under traditional Marxist intellectuals. Also neo-Marxist didn't believe that the poor would overthrow the rich and form a communist society. They believed that class-conflict was not only universal but nearly eternal and to break out of it we would have to practically erase everything and start over again since the bases of classes wasn't just in economics and the material world but was well-rooted and supported by our culture, language and morals.

The difference between the two schools can be better understood with an example. Marx said that the Bourgeois remain in power basically because of two things: the inheritance laws (which make sure that the rich person's son is also rich and they remain in control of the factors of production) and ideologies (which continue the ideas on in the next generation so that the poor accept their poor status because it is "the way of things". Marx was also very anti-religion since he believed that some religions promote this fatalistic approach. He believed that the rich also use religion to control the poor-which the church actually did back in 18th century Europe). The neo-Marxist school extends this idea and says that there are several more sociological factors which continue capitalism namely; education systems (like how the rich students can easily get into prestigious schools and hence have a better chance of making more money-thus continuing the cycle), socialization (the children of the rich are taught to think outside of the box. They are given everything since birth and hence this gives them the confidence needed to be successful in life while the poor kids usually lack confidence and are more self-conscious. This can impede their opportunities and growth in professional careers. This idea was later called the "culture of poverty". I'd suggest you look it up. Also another example of how socialization affects classes is how we are just wired to distrust the poor and trust people who are well-dressed and look successful at first glance. Why is this? The rich person could be a psychopath but we make assumptions and these assumptions are made purely on ideas we formed as result of our early socialization) and lastly culture (we somehow consider only the rich to be "cultured" when that makes no sense. Culture does not JUST include classical music, theater plays and fine arts. And culture doesn't belong to anyone but somehow over the years the idea has taken root that the poor are somehow "less" that they are lacking in the cultural stage. And these sort of ideas are what enable the rich to maintain their stronghold over the poor).

The above is just one of the key points in how these two schools differ. There are many more differences and I would like to point out that there are several more sub-branches of Neo-Marxism and many of them don't even agree with each other. Also note here that Neo-Marxist usually hold left-wing political views so I would suggest you read up upon those as well to better understand the practical use of their theories.

I am an economist by education so I never really learnt much about Neo-Marxism since its more of a sociological theory and builds on Max Weber's ideas of class and privilege, but I hope this gave you a good starting point.

a really good explanation!