As Pace Nielson Nielsen already posted, the strength of quiver theory is to provide easy examples and counterexamples.
The first applications are of course inside representation theory and ring theory, because Gabriel's Theorem states, that if you have a property of a finite dimensional algebra over an algebraically closed field that can be detected in the module category, then it suffices to look at path algebras of quivers (with relations). For example for proving that an algebra is wild, it suffices to find a subquiver (with relations) that is known to be wild; and there are several lists of such quivers. This is useful in representation theory of Lie algebras and finite groups.
There is the connection with Lie theory (and other things that can be classified via Dynkin diagrams) via the Hall algebra.
Cluster theory, which is an advanced topic in representation theory of quivers, has applications in geometry.
If you are given an algebra, I think it is a natural question is, whether it is possible to classify all the indecomposable representations. If this is possible you can work on a parametrization and understand better the module category by working with the classification. The tame-wild dichotomy helps you there, it answers the question if it is possible. If an algebra is wild, it is not possible to classify all representations. You have to ask other questions.
