Neo-classical has two meanings I’d say:

1) A disparaging term by people who don’t understand economics to attack economists with 😉
2) The neo-classical synthesis which brought together insights from the classical and Keynesian schools post WW2 to come up with a tractable way of discussing macroeconomics.

Of course, when the neo-classical synthesis broke down we had a whole bunch of stuff – post-Keynesians were there, there was a New Keynesian school that is different from the current one, and of course New Classicals.

While the “New Keynesians” (I’ve heard them called neo-Keynesians before) discussed the idea of microfoundations for pricing issues and competition issues, New Classicals developed the Real Business Cycle framework – essentially they used the idea of intertemporal maximisation and rational expectations to form a tractable way of describing the macroeconomy.

Eventually, people started putting the New Keynesian and New Classical ideas together, and we got the modern New Keynesian style model – which is a intertemporal model with rational expectations, and a bunch of nominal rigidities (or other things – more and more game theory is getting involved). I’ve also heard this called the New Keynesian synthesis before – and a bunch of other names.

New Keynesian is the predominant school at the moment, and as a result people that don’t understand economics will call it “neo-classical” as it is mainstream economics – I believe that is the reason.