The major difference here is that the Keynesian model believes that government involvement is necessary, at least when the economy is in a deep recession. The classical model believes that the economy is self-correcting and that it will always be able to return to its equilibrium without government intervention.
The Keynesian model argues that the aggregate supply curve is flat (at least in a deep recession). This means that the government will need to move the aggregate demand curve by spending more money. By doing this, the government increases GDP without increasing the price level.
The classical model argues for a vertical AS curve. It says that government actions that move the AD curve will only affect the price level, not GDP. Therefore, it says, the government should stay out of the way and let the economy correct itself. The economy will get back, in the long run, to full employment without inflation.