mathematical physics - Causality and natural modeling of physical systems using integral forms

I posed a closely related question here but it received a tumbleweeds award. So I thought I would post it from a different angle to see if I can illicit at least some thoughtful comments if not answers.

The modeling of many physical systems utilize the mathematical tools of calculus, by writing the relationship of physical quantities in the form of differential equations.

Considering time dependent operations of integration and differentiation, the dynamics of a physical system may be expressed in terms of one form or the other. A good example are the Maxwell Equations which are often written in both differential and integral forms.

Integral forms tend to express where the system has been up to where the system is at present while differential forms tend to express where a system is now and where it will be in the near future. So the two forms tend to imply a sense of causality.

So this brings me to my question. Since we tend to observe a causal universe (at least at a macroscopic level) are integral forms a more natural approach to modeling systems?

I'm using the word 'natural' in the sense that the nature of the universe tends to work one way vs another. In this case I'm saying nature tends to integrate rather than differentiate to propagate change. We can write our equations in differential form, solve them and predict, and they are useful tools. But isn't mother nature's path one of integration?

I tend to believe this is so by my experience in simulating systems. Simulating systems in an integral form rather than differential form always seems to lead to better results.