Volatility is a very important concept in finance for the simple reason that it matters how you get there. Let's say you need to get from Chicago to Indianapolis. You have two options to get there, you could drive your nice luxury SUV with air conditioning and cruise control or you could hop into a railcar as a stowaway. Its summer and the railcar is 110 degrees inside and you're so hot after an hour that you feel like jumping out at a stop only halfway there. You get slammed into the side of the railcar and injure your shoulder and now you can't use your right arm and the pain is killing you. You now want to jump out and do anything to help alleviate the pain. You decide to tough it all out and arrive in Indianapolis three hours later, the same amount of time it took to drive. Both options resulted in the same outcome, getting from point A to point B but the railcar option was extremely uncomfortable and you considered not making the whole trip a few times.
The same goes for stock returns, the degree of discomfort experienced on the trip to year end stock market gains is expressed by standard deviation and volatility. Excessive volatility will make you want to sell out of a stock or portfolio. It can make the ride extremely uncomfortable. A rational investor, even one who claims he's in it for the long term will be tested to sell out at a bottom with too much volatility.
With volatility and standard deviation, you can calculate one of the most critical metrics of any portfolio which is risk adjusted return. Formulas like the Sharpe and Sortino ratios help quantify the risk adjusted returns of a stock or portfolio.
Aside from options pricing, this is the best application of volatility and standard deviation to portfolio construction and financial econometrics.