The main shortcoming of delta hedging is that a delta hedge requires frequent re-balancing. When ever the price of a security changes, so does the delta of any derivative based on it. When the delta changes significantly, the composition of any delta hedged portfolio will need to be changed.
The greater the change in the delta, the greater the change that is needed in a delta hedged portfolio. Therefore the amount of re-balancing needed can be reduced by reducing the amount by which the delta changes for a price movement — in other words by reducing the rate of change of the delta, which we call the gamma.
A gamma hedged, or gamma neutral, portfolio will need to be re-balanced less than one what is only delta hedged. It is dynamic hedging and therefore needs re-balancing, especially if there are large price movements, because the gamma, like the delta, changes with underlying price. However, it needs smaller adjustments than a pure delta hedge.
Gamma hedging is also more complex because it requires holding two derivatives to hedge a each holding of a single security.