If one has a deterministic model of physics, why is causality so important?
Let's work in a fixed frame. Suppose that event A in the future causes event B, which happens before event A. Now, given the conditions at some time $t_0$ before both events (or at the time of event B), one can predict what will happen in the future, hence one already knows that event A will happen.
Therefore, there's no event that happens after B that causes B and that we didn't already know about at the time of event B. Rather, we could say that because event A is caused by the conditions at time $t_0$, event B is in essence caused by the conditions at time $t_0$ and not by event A (in the sense that it can be explained completely in terms of what happens at time $t_0$). Therefore, if we have a theory that says that event A can cause event B, we could re-explain the exact same theory by considering the mechanism that allows A to cause B as simply part of the laws which allow us to determine all future events from the initial conditions at time $t_0$. Maybe, there's some mathematical aspect of those laws, some term in the equations, which we would philosophically like to think of as 'event A causing event B.' But physically, causality is already in this sense satisfied under a deterministic model.
I came to wonder about this as a result of classical electrodynamics (without any other forces or gravity). If we know the electromagnetic field at a given point in time in addition to its time derivative, as well as the position and velocity of all charge and mass, we can use Maxwell's equations and the Lorentz force law to find out what will happen at all future events.
The implication of this is that in special relativity, we shouldn't really have to worry so much about information traveling faster than the speed of light (something we impose on a system because otherwise it would violate the law of causality after applying the Lorentz transformations). It does so happen that the current classical special theory of relativity works nicely mathematically and also satisfies that principle, so we have no reason to reject it, but I don't see why it's so important for information not to travel faster than the speed of light.
One further idea: We are assuming that we can choose some time $t_0$ before all events caused by A, but what if A causes events arbitrarily far into the past. One idea is that we could in some sense consider the limit of the state of the universe as $t$ approaches $-\infty$ as our initial conditions.
Maybe the independence of causality and determinism could make sense in a universe with 'divine intervention' of sorts, where we could suddenly decide to intervene at any point and break the deterministic model. For example, suppose we could suddenly give an electron in the universe a mysterious push, or we could even magically make an electron appear or disappear. Then if the laws of physics allowed that divine push to influence what happened in the past, that would violate causality. But that situation would not be deterministic.
Best Answer
Causality means that for any two events A,B, there has to exist an ordering that says whether A can influence B or B can influence A, and in the normal examples with "time", the ordering is the condition $$ t(A) < t(B). $$ If the condition above is satisfied (i.e. if A precedes B), then A may influence B.
An ordering - a transitive relation - has to exist in order to avoid logical contradictions. If the relationship were not transitive, for example, it would be possible to find triplets of events A,B,C such that A influences B, B influences C, C influences A. That would be a "closed time-like curve" and it would lead to logical inconsistencies because in general, there would be no way to choose the outcomes of the events A,B,C so that all the three implications are preserved.
Those contradictions are avoided in any causal theory because the outcome of event A (in a causal and deterministic theory, to be specific) is never calculated from conditions at event B if A is the cause of B (if A precedes B). It's the other way around. Causality makes it clear which data are "inputs" and which data are their "outputs", so because of this orientation, there can't be any contradiction.
In a geometric setup, the comparison of a coordinate associated with the events is the only way how to produce ordering as a relationship. We call this coordinate "time".
In special (and similarly general) relativity, the condition for A to be able to influence B becomes sharper - $t(A)<t(B)$ has to hold in all reference frames which means that B has to belong to the future light cone of A.