Cost-push inflation generally has a bigger impact on the economy. This is because it is much more likely to cause recessions or even stagflation.
Cost-push inflation tends to cause a reduction in aggregate supply. As the cost of producing goods goes up, the amount that producers are willing to supply at any given price level goes down. Graphically, this can be represented by a movement of the aggregate supply curve to the left. When this happens, all other things being equal, the price level increases and, at the same time, the Real GDP goes down. This is much worse for an economy than the situation with demand-pull inflation, where both CPI and RGDP go up.