Short run growth is an increase in AD, meaning any one of the compenants in aggregate demand increases. This can occur if people have a change to their disposable income, for example if taxation is reduced people will have an increase in dispoable income and may spend more. This would have the effect of increase of consumption thus causing AD to increase. Therefore, all short run growth will cause an increase AD, moving the economy towards a higher price level thus causing inflation.
Long run growth, is an increase in all or any of the factors of production causing an increase in aggreate supply, as it's a change in the potenial growth of the economy. This is the fundemental diffrence between short run and long run, short run is actual growth, while long run is potenial growth. Long run growth alows for future growth as it expands the PPC of the economy. There are two ways to increasing aggregate supply (which is long run growth) and that is via either by increasing the factors of production, or impoving them by making them more efficient e.g. investing in human capital to make workers more efficient will increase the economic capacity of the economy.
Therefore, the difference between short run and long run is that short run shifts the AD curve causing an increase in inflation due to higher price levels, whilst long run allows for the economy to have higher AD in the future as it allows for higher levels of AD before reaching full employment. So both short run growth and long run growth are necessary for a growing economy.