I think Stochastic Calculus is important to know because it really lays out the foundation for pricing any derivative payoff, as well as the various risks associated with such payoffs. You can price stock options, futures, forwards, FX, interest rate derivatives, and pretty much anything using Stochastic Calculus - I think what it does at least from a practical standpoint is it unites the (usually very disparate) worlds of equities, futures and commodities, and fixed income into one very solid framework which then yield closed form solutions and PDE's for derivative prices that you can solve numerically.
Whether it is still applicable in today's world is an entirely separate matter altogether.
Whether it is still applicable in today's world is an entirely separate matter altogether.