Your question is a bit hard to answer, because it does not make sense to compare causal calculus (a set of inference rules to manipulate causal models) to differential equations (a type of model). So first let's make clear what the causal calculus (or the do-calculus) is.
The do-calculus is a set of three inference rules that help manipulating probabilistic sentences involving interventions in structural causal models. The three rules determine:
when you can insert or delete observations in probability statements (rule 1);
when you can exchange action for observation in probability statements (rule 2); and,
when you can insert or delete actions in probability statements (rule 3).
The conditions for applications of these rules depend on the assumptions of your causal model, in this case encoded by the DAG. The goal of having these rules is to "massage" the query, which is usually an interventional expression, into a purely observational expression (which can then be computed from observational data). The problem of deciding whether your causal assumptions are enough to express the causal query only in terms of available data is what we call the identification problem.
Thus, it doesn't make a lot of sense to ask what is the difference of causal calculus and differential equations, since they are not even in the same class of objects. What might make sense, though, is to ask what is the difference between structural causal models and differential equations. And here I think we can make two distinctions regarding: (i) the level of representation of the phenomenon; and, (ii) what types of problems each literature is trying to solve, and what type of tools they have developed.
The short answer for the first question is that, usually, the models are describing the phenomenon at different levels of abstraction (DE describe continuous changes, whereas SCM equilibrium states). The literature bridging both representations is fairly recent, you may check here, here and here for more discussion.
Regarding the second question, note the causal modeling literature usually is trying to answer questions of the type, "what can we learn from a nonparametric partial specification of a causal model (encoded by a DAG, for instance) and the data? Is that sufficient to answer your causal query?" This is a different problem than, say, assuming to have a known system of differential equations for which you only need to estimate some parameters from data, but for which there's no identification problem involved.