Continuous time has a so-called elegance, but it is rarely correct. Most Q-measure people rarely care about correctness anyway, since they usually don't root their models in statistics. With no goodness of fit measures, continuous time models are elegant theory.
In general, we also see that most ex-ante hedges are rarely good, ex-post. They have large elements of directionality. There are lots of minor alterations, and even kluges (e.g., hedging delta by using the 'smile'). Even simple things like calibrating implied vol is technically wrong (recall the P-measure dynamic in Black-Scholes uses the same vol-i.e., the change of measure doesn't change the vol, so technically it must be the same as given by the historical dynamic, the DGP-in a Black-Scholes world, there is no implied-realized premium!). Of the many standard methods for hedging swaptions I have seen and used and backtested, it's pretty clear none are great.
In some ways you could say that continuous time finance with its beautiful formulas and elegant equations is merely a method for splining (calibrate to 4 points on the smile, and infer all others). But as I say rarely do Q-measure types care about the reality of P-measure. (exception: the failed attempt of the Macro-Affine community). Model Validation people try to do this correctly, but as far as I can tell, their methods are not altogether satisfactory.
Continuous time finance gives us some nice formulas and rules of thumb. But the world around us can be modelled more effectively for the most part in discrete time. Moreover, the dynamics are far richer in discrete time. Autocorrelation, seasonality, long-lag lengths--all of these phenomena are impossible to fit into SDEs.
One must know continuous time mathematics to understand and get an intuition about optionality and nonlinear payoffs. But until academics and practitioners start using delay-differential stochastic differential equations (used in areas such as signal processing/electrical engineering, and other physical sciences) in finance, we can model many more interesting phenomena in discrete time than we can in continuous time.