My intuition says that both approaches, discrete time models and continuous time models will be models (i.e. approximations) of reality. Therefore it should be possible to develop useful models in both domains.
Continuous time models have more mathematical elegance and can therefore probably bring more mathematical machinery to bear on the problem which presumably helps with deriving analytical solutions and asymptotic limits.
Discrete models more easily correspond to observed data and measurements and are easier to simulate on computers.
I have been told that it is possible to discretise continuous time models and vice-versa but care has to be taken when performing this transformation. Could you please highlight what the common pitfalls are, in particular when modelling asset prices? If there are other differences in the dynamics between the two approaches (for example possibly something like non-linearities, chaos, ... in one and not the other) then I would like to know about that as well.