For a call option, we know that the vega is the derivative of the price wrt to the volatility. However the volatility, in that context, actually refers to the implied volatility of the specific call contract at this moment, which is inferred from the option price. So, knowing this, what is the purpose of the vega ? We are essentially saying "if the implied vol rises by 1%, the price rises by X", but the cause that will trigger a change in the implied vol is the price of the option itself.
Think of it this way, the change in option premium is roughly the sum of changes due to delta, gamma, theta, rho, and vega. Now for a long dated option, gamma and theta don't change dramatically day to day, and let's say you have delta hedged your option. Then your P&L is mostly just driven by changes in implied vol – i.e., vega. So vega matters a great deal when you are trying to manage risks of an option book – you can isolate different sources of risk (due to underlying, due to time, due to implied vol) and hedges them accordingly. Similarly, there are also strategies to trade (almost) purely on implied vol – if you carefully hedge out all the other risks, then your P&L will be completely driven by vega. This, of course, is because of the one-to-one mapping between implied vol and option pmreium, just like you said.
I thought about it more. If you are familiar with bonds, I think they provide a good analogy. As you may know, bond price and bond yields also form 1-to-1 mapping, with "DV01" or "Duration" measuring the sensitivity of bond price to yield changes. The value of "DV01" (or duration) is that they provide a consistent framework for measuring interest rate risks. Two bonds might be trading at very different prices (e.g., if one has much higher coupon than the other), but they may have similar duration, so the two have similar risks. Simply put, bond prices are not particularly meaningful, but bond yields are more comparable (that's why we plot the "yield" curve, not bond "price" curve...), and yield-based risk metrics are also more useful.
Options are the same. Prices are not particularly meaningful, while implied vols are more comparable across different options, allowing for easier comparison, hedging, and P&L attribution. As I mentioned above, if you can carefully manage other risks, it's possible to trade vega directly. In fact, we frequently look at stuff like spreads of two implied vols and trade on relative value in this way.
As an option trader you may well prefer model and forecast volatility rather than the specific price dynamics of the stock and option contract. This is a straight-forward consequence of the Black-and-Scholes framework (*). If you own an option, knowing vega allows you compute the expected PnL on your position and manage risks with respect to your realized volatility forecasts.
(*) As a matter of fact, the real-world stock-price drift is not an input of the B&S formula.