They are not the same, but they are related.
Gamma is sensitivity to realized volatility. Vega is sensitivity to implied volatility. Vanilla options are always long gamma and long vega, so they are "long vol" and saying "I am a buyer of vol/gamma/vega" means that you are taking a position that benefits from a rise in volatility (either realized or implied).
Although vanilla options are long both gamma and vega, they are generally long in different amounts. Near expiry options have more gamma, and far expiry options have more vega. That means you can construct a long gamma/vega flat portfolio by buying short-term options and hedging the vega with a short position in long-term options, or you can construct a gamma flat/long vega portfolio buy buying long-term options and hedging the gamma with short-term options. Each of these portfolios would be "long vol" but one is only long gamma, and the other is only long vega.