The problem is that what some mean when they say "volatility" is BS implied vol from an option price. What some others mean when they say "volatility" is some diffusion parameter from a drift diffusion model (with or without jumps). These are the same value in the log normal model of stock prices but different for many other models including those with jumps. Therefore, a model with jumps would have a higher option implied volatility as mention by @Farahvartish in the comments, but it may have its own diffusion parameter which could be called volatility that would be lower than the option implied volatility.
To answer your question as to why anyone would care about better modelling, it comes down to arbitrage and hedging portfolios. If I can better price and hedge an option position than you can, I can make money from inconsistent pricing in the market.