The main difference is that the classical model of economics believes in a self-correcting economy while Keynes argued that the government will sometimes need to intervene, particularly to get a country out of a recession.
According to the classical model, the economy will always return to full employment on its own in the long run. Declines in aggregate demand, for example, will lead to price drops which will return the economy to its previous equilibrium at full employment. Keynes argued that this does not actually happen. He argued that the government had to intervene in such situations, increasing aggregate demand via government spending.