I'm not an expert, though I'm studying stochastic calculus right now in one of my classes. The Black-Scholes price gives an exact solution to European Call option prices, subject to fixed risk free rates, constant volatilities, and other assumptions. It can be used as one tool (out of many) to hedge a portfolio, such that, at any time, regardless of the change in price of the underlying, the change in value of the portfolio will be positive, subject to other assumptions like, for example, the option price vs underlying price curve doesn't straighten out (it maintains convexity as time goes on). So, you can use the Black-Scholes equation to create a delta-neutral portfolio, in theory.
In actuality, it's much much more difficult than this. Everyone already knows the pricing formulae, so in theory, all of the profit has been arbitraged away. Any advantage will last for a few seconds (imagine 100 other people just as smart or smarter than you trying to find a pricing mistake). Secondly, even if an investor did have a strategy, the execution of that strategy requires considerable skill. Simply buying or selling a stock incurs price slippage, trading costs, feedback, etc. There are entire departments in the banks devoted to trade execution alone.