Marx and Engels view history primarily as the story of class struggle. The bourgeois class owns the means of production and employs the proletarian class to do the work. The interests of these two classes directly conflict, and the bourgeoisie will always exploit the proletariat until it is overthrown, replacing capitalism with communism and creating the classless society.
Bauman views class as a modern concept, meaning that history cannot be the story of one class struggling against another. His point is not that inequality is new, but that the idea of class is a modern hybrid, based on social status, financial power, education, and various other factors. Feudal societies were based on what was essentially a caste system, in which you were born with a status you could not lose. The son of a lord would be a lord, no matter what hardships befell him or how stupid and wasteful he was. The son of a serf was similarly forced to occupy the same position as his father.
In the modern era, class is largely determined not by birth, but by the type of work you do and how much money you have. The Marxist class struggle, according to Bauman, demands a certain degree of social mobility and was born out of the Industrial Revolution and the factory system rather than being an essential feature of history.