It assumes that the underlying model follows a Gaussian distribution but as Mandelbrot showed a Lévy distribution is a better model.

Black-Scholes is a name for a formula that was around before Black and Scholes published. Beforehand it was simply a heuristic used by traders. Those traders also did scale a few parameters around in a way that a normal distribution wouldn't allow. Black-Scholes then went and proved the formula correct for a Gaussian distribution based on advanced math.

After Black-Scholes got a "nobel prize" people stated to believe that the formula is actually measuring real risk and betting accordingly. Betting like that is benefitial for traders who make bonuses when they win but who don't suffer that much if they lose all the money they bet. Or a government bails you out when you lose all your money.

The problem with Levy distributions is that they have a parameter c that you can't simply estimate by having a random sample in the way you can estimate all the parameters of Gaussian distribution if you have a big enough sample.

*I'm no expert on the subject but the above is my understanding from reading Taleb and other reading.