Einstein-aether theory is a theory of gravity with the local Lorentz violation. In addition to the metric tensor, it contains a unit timelike vector field, called aether $u^a$. Because of the constraint $u^au_a+1=0$, $u^a$ never vanishes, so people say that the local Lorentz symmetry is dynamically violated (see gr-qc/0007031).
Now, let's ignore the metric part, that is, we work in the Minkowski spacetime and call this theory Aether theory. As you can see, the aether action is perfectly Lorentz invariant no matter whether you carry out an active or a passive Lorentz transformation. Why is Einstein-aether theory violating Lorentz symmetry?
In Einstein-Maxwell theory, or in the flat spacetime version - Maxwell theory, Lorentz invariance perfectly holds, too. The 4-potentail $A^a$ is not constrained to be a unit timelike vector. Maxwell theory is definitely Lorentz invariant, no matter whether $A^a$ vanishes or not. Even if $A^a\ne0$ in some cases, Maxwell theory is Lorentz invariant as textbooks tell us.
Whether a theory is Lorentz invariant or not does not depend on whether there exists a particular solution with a nonvanishing four vector, like the $A^a$ in the last sentence of the previous paragraph. It is only determined by whether the equations of motion preserve the same form under the active Lorentz transformation. This is actually Einstein's special relativity principle. In this sense, Maxwell theory is Lorentz invariant, so is Aether theory!
So it seems that the only difference between Aether theory and Maxwell theory is that the aether field is nonvanishing, while $A^a$ vanishes in some cases. However, Aether theory is Lorentz violating, but Maxwell theory is not. Why?