The main advantage of chiral algebras over vertex algebras is that they admit "very functorial" definitions, and this helps more general concepts and constructions appear naturally. The usual examples involve factorization spaces like the Beilinson-Drinfeld Grassmannian, and applications to the Geometric Langlands program. Another example is the concept of chiral homology, which can be viewed as derived version of a coinvariant construction, but has taken on a new life as factorization homology in the theory of extended TQFTs.
I disagree with the claim that the definition of "chiral algebra" is shorter than that of "vertex algebra". You really need to build up a substantial body of theory even to describe the chiral pseudo-tensor structure on D-modules. Vertex algebras are fundamentally vector spaces with a funny looking multiplication structure.
If you are trying to decide whether you want to invest more time in learning about chiral algebras versus vertex algebras, it may be worth your time to ask which notion has yielded substantial results that interest you. Chiral algebras have been available to the public for about 20 years (since Gaitsgory's IAS notes), and vertex algebras have been around for about 30. That is, neither notion is particularly new.