SAT is about a very special case of logic, namely propositional logic, where every variable stands for exactly one of only two possible truth values. Finding satisfying assignments in propositional logic is indeed already a computationally hard task, and it is good that we have efficient SAT solvers for such use cases, allowing us to solve a large variety of combinatorial problems nowadays.

However, Prolog lets you tackle not only propositional logic, but tasks that go far beyond this, belonging to a logic called classical first order logic, of which propositional logic is only a subset. In first order logic, we reason about predicates between terms, and this lets you tackle much more complex tasks, far beyond what SAT/SMT solvers can solve for you.

In fact, first order logic is so powerful that it lets you describe everything you can in principle perform with a computer. It lets you describe how a compiler works, for example. Or how numerical integration works. And all other computations you have ever seen computers perform.

In short, Prolog is a programming language, and can in fact even be used to implement a SAT solver, which is impossible to do with just a SAT solver. Many Prolog implementations even ship with a SAT solver as one of their libraries.

There are also even higher-order constructs in Prolog, such as predicates that let you invoke other predicates, making programming in Prolog very expressive and convenient.