What is the difference between the domain of influence and the domain of dependence?

I'm a physicist, so this answer will draw inspiration from how this concepts arise in Relativity.

Quick answer: the domain of dependence of a set $S$ is the collection of points $p$ such that $u|_S$ depends on $u(p)$, while the domain of influence of $S$ is the collection of points such that $u|_S$ can influence $u(p)$.

As you mentioned, the domain of dependence of some set $S$ is the collection of points that are completely determined by initial conditions given at $S$. In the particular case of the wave equation in usual flat space(time), it has the form you mentioned.

The domain of influence (in General Relativity you can also find the terms chronological future/past) of $S$ on the other hand is the collection of points that somehow are influenced by $S$. If a point is on $I(S)$, it ''feels'' the influence of what happens at $S$. If I change the initial condition for $u(x_0, t_0)$ it will, in general, affect the value of $u$ on $I(S)$, but points outside of $I(S)$ won't notice it: they are too far for that information to reach them.

Pick $(x_0,t_0) = (0,0)$ for example and set $c=1$ for simplicity. The point $(2,1)$ is too far from $(0,0)$ to be influenced by the initial condition at $(0,0)$. Physically speaking, if I switch on a light bulb at position $x = 0$ at time $t = 0$, someone at position $x=2$ won't have seen the light yet at $t=1$: light didn't have enough time to reach them yet. On the other hand, light will have reach someone at $(x,t) = (2,3)$, so $(2,3) \in I((0,0))$.

In this analogy, the (future) domain of influence of $(0,0)$ is he collection of events (spacetime points) which can see the light bulb has been switched on. The domain of dependence of $(0,0)$ is only $(0,0)$ (intuitively, this is because other points could be lighted by light bulbs in other places, so the initial condition at $(0,0)$ alone is not sufficient to determine the function's value at them).