Let $(M,\omega)$ be a symplectic manifold, and let $\mathscr{J}$ be the set of compatible almost complex structures on $M$.Finally let $A \in H^2(M,\mathbb{Z})$. Then we can consider the moduli space of parametrised J-holomorphic curves defined as follows.

$\mathscr{M}(A,J) = \{ (u,J)~|~ J\in \mathscr{J} ~ \text{ and u is J-holomorphic curve} ~\text{such that}~[u]= A\}$.We know that the automorphism group $G=\text{PSL}(2,\mathbb{C})$ of the sphere acts on $\mathscr{M}(A,J)$.

I read in a couple of papers the comments that $\mathscr{M}(A,J)/G$ can be compactified by adding cusp curves i.e connected unions of possible multi-covered J-holomorphic curves, but the "right" notion for the compactification of this space is by considering the finer notion of stable maps but this "simpler" notion suffices for all purposes in the paper. 

Could someone please explain why compactification by stable curves is "better" and under what conditions can we work with just cusp curves?