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The fractal model of markets never caught on because it was too hard to make it practical for the kind of thing that practitioners cared about — i.e. the pricing and hedging of large derivatives books. We developed more practical models that solved the problem in a different way,
largely local volatility and later stochastic volatility models which explicitly build in time-varying volatility and hence generate fat tails and heteroskedasticity. Like all models they are wrong, but unlike the fractal model, they are useful.
That’s all I have to say about it really. Mandelbrot was not wrong, in fact he was ahead of his time — but his research led in an unproductive direction. The fetishisation of his ideas that has cropped up in the last 10-15 years is weird and I attribute it mostly to people who
have read “fooled by randomness” or “the black swan” once and largely failed to understand them 
I feel like @EmanuelDerman would have a much better take on this than I do!
any resources from which an interested finance student can learn more about this particular topic (modern volatility models)
Have you took a look at Taleb's latest ? "Statistical Consequences of Fat Tails". Wanted to know your take on it.
@KeithMcCullough @HedgeyeDDale I would be interested to hear your thoughts on these comments as I am not proficient enough to reply but I know your process (which I can testify works well) uses Mandelbrot's findings as part of the building blocks?
LV model (dupire) doesn't generate fat tails. It just matches prices of European options SV models generate smile in the imp vol surface, very generous to call that fat tails, but it is allowed Fractal models are becoming more popular, see rough volatility by gatheral
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