Keynesian Economics and Classical Economics

It might be quite a little old question, but what is the difference between Keynesian Economics and Classical (Walrasian) Economics?

Now in economics there's no big disagreement about the coexistence of Keynesian and Classical economics, but many economists explain about Keynesian economics from the viewpoint of Classical economics and they think that economics should be based on Classical economics.

The difference is not related to some ideology, but to point of view: Keynesian economics focuses on the short-run and demand-side economy while Classical one on the long-run and supply-side economy.

Of course, both sides of economics agree that competitive households and firm think at margin when making an optimal decision and that competitive economy is Pareto-optimum.

However, they think differently when they look at the whole economy: Keynesians think that the whole economy should fail to be composed, what is called the 'fallacy of composition', and some little shock (tax policy change or so) can be very big impact on it. In contrast, Classical economists think that the whole economy should be consistently composed by each optimal decision making and thus that some little shock can be as little as itself on the whole economy.

Classical economists admires 'Neutrality Theorem', while Keynesian economists 'fallacy of composition'. The former says that changes in monetary expansion or government expenditure have almost no impact on the whole economy because each person makes every decision as looking ahead in the future, while the latter says the opposite because each person doesn't look forward.

Which idea is right? The point is the view each side of economics puts weight on. Some problems are though to be better analyzed from the viewpoint of Keynesian economics while others are not. Which side should be taken depends on what economists think as of importance.