The first key advantage of discrete time is that all economic and financial data are ultimately only available in discrete time. (This might be surprising for the case of finance, but it should be noted that the entire premise of the profitable high frequency trading industry rests upon the observation that financial transactions are not instantaneous. Continuous time models are used in mathematical finance, but these should be interpreted as approximations of the true system.) A continuous time model is therefore one step removed from the data, and we would have to be cautious translating properties that appear only in continuous time series.
Even comparing a discrete time model to data is always going to be a difficult process in practice. For example, how do we treat monthly data in a model that evolves quarterly? We are always going to lose information (or forced to insert information) as we change data frequencies. Furthermore, it is difficult to align data that released with a variable lag to the calendar dates that they represent.
A second key advantage of discrete time is the simplicity of treatment, particularly if random variables are involved. As soon as we introduce randomness, it is incorrect to assume that the derivatives of any variables exist. To what extent solutions exist, they are defined in terms of Lebesgue integrals, a mathematical area that is not particularly well known. Almost all the work in analysis proofs would involve extremely obscure corner cases. (“What happens if government spending is $20 if t is rational, $0 otherwise?”) It is one thing to define continuous time models where the components are passive resistances and capacitances that obey simple laws of physics; the interactions created by entities reacting in real time to inputs creates the possibility of highly pathological outcomes.
A related issue is the question of time delays. Within a discrete time model, a time delay is straightforward: we add a new state variable that is the original variable from the previous period. In continuous time, the amount of information contained within any non-zero interval is theoretically infinite. (For example, we could theoretically encode all human knowledge into a signal that lasts less than one microsecond. In practice, information channels have finite bandwidth, so we do not see this effect.) In order to model a time delay, we have an infinite dimensional system. Statements of mathematical results (such as stability theorems) we have available for infinite dimensional nonlinear systems would comprise a very small book.
Finally, accounting is unusual within a continuous time system. We are no longer doing familiar accounting with stocks and flows that can be related with basic arithmetic. We instead would have to define all accounting relationships as stocks being the Lebesgue integral of flows. Such an environment is much less intuitive, and more prone to error. Furthermore, there is no clean way to model events that cause discrete jumps in stock variables, without invoking the Dirac Delta Function. This so-called function is not actually defined as a time variable, and so it is difficult to relate it to system behaviour that is defined as mathematical operations on time series.
The only real cost to discrete time analysis is that some of the more easily understood stability results (such as Lyapunov functions) are lost. However, it is possible to define the discrete time equivalents, and the general lack of computational tractability of such results for high dimensional systems makes the loss of this theory not practically significant.