Neo-classical synthesis is a synthesis of classical model and Keynesian model. In brief, it states that Keynesian model is correct in the short run whereas the classical analysis is correct in the long run.
Let's consider a concrete instance. According to Keynesian model, an increase in G will increase Y and decrease unemployment. In the classical model, an increase in G would have no effect at all on Y and unemployment. In neo-classical synthesis, an increase in G will create a temporary increase in Y however Y will return to its original value after some time.
To justify neo-classical synthesis, it's helpful to identify the problem with classical model in the short run and problem with the Keynesian model in the long run. As for classical model in the short run, we figured out that within this model, it's difficult to describe deep recessions with high involuntary unemployment. In the long run, it's more reasonable to believe that the economy can get out of the recession by itself. Problem with the Keynesian model in the long run, as we would see, is assumption of a stable Phillips curve.