Classical economics is the original school of economic thought first developed by Adam Smith (Wealth of Nations, 1777). It lay out many principles of economics that were then to be built on by the neo-classical school. The neo-classical school is the current dominant school of microeconomic thought. Classical economics looks at value and distribution theory: goods and services are valued at the cost it took to create them (labour theory of value).
Neo-classical economics as instead of believing that prices are determined from the cost to create the good, it focuses on the value consumers and firms place on the good. As such, it looks at interactions between supply and demand in a market as being the determinant of price. Classical economics was still very close to social analysis whereas Neo-classical economics introduced many concepts that made the discipline more "scientific".