There appears to be "classical", "new classical", and "neo classical" economics.
Classical considers that competition leads to the most efficient use of resources and that supply and demand are regulated through market forces.
Neo classical, like classical, considers all the aforementioned points, but differs in considering that savings drives investment, not vice-versa,
New classical considers the circulating money supply, tax cuts, and minimal government fiscal regulation to be key economic factors.
Yes, the key development (rather than difference) between new and traditional Keynesian approaches to economics is that there is a shift from macro to microeconomics in its justification of its economic theories and position. There is no central difference, rather just a shift in the means used to support it.
The main differences between these two sets of "new" and "traditional" views is that the new ones are more interested in microeconomics while the traditional ones are more macro. The new versions do not really disagree with the old ones. Instead, they are trying to bring microeconomic rigor to bear to explain why and how the old versions are right.