Convergence to cusp curves is the original compactification by Gromov, whereas convergence to stable maps is the compactification by Kontsevich (a cusp curve corresponds to the image of a stable map). The latter is more accurate if you want to correctly model the topology of the moduli space, so as to build ``virtual (pseudo)cycles'' and use the full power of the Gromov-Witten invariants (this being in the realm of algebraic topology). I believe an explanation is in Chapters 5+7 of McDuff-Salamon's big book. (There they mention that if your manifold is semi-positive then the usual Gromov-Witten invariant can be built without the finer notion of stable maps.)
Think of it this way: You could also take the one-point compactification, but that probably won't tell you a lot.

Convergence to cusp curves is the original compactification by Gromov, whereas convergence to stable maps is the compactification by Kontsevich (a cusp curve corresponds to the image of a stable map). The latter is more accurate if you want to correctly model the topology of the moduli space (with a fixed number of marked points), so as to build ``virtual (pseudo)cycles'' associated with evaluation maps on the moduli spaces and hence use the full power of the Gromov-Witten invariants. I believe an explanation is in Chapters 5+7 of McDuff-Salamon's big book. (There they mention that if your manifold is semi-positive then the usual Gromov-Witten invariant can be built without the finer notion of stable maps.)
Think of it this way: You could also take the one-point compactification, but that probably won't tell you a lot.

Convergence to cusp curves is the original compactification by Gromov, whereas convergence to stable maps is the compactification by Kontsevich (a cusp curve corresponds to the image of a stable map). The latter is more accurate if you want to correctly model the topology of the moduli space, so as to build ``virtual (pseudo)cycles'' and use the full power of the Gromov-Witten invariants, such as applying homological algebra. I believe an explanation is in Chapters 5+7 of McDuff-Salamon's big book. (There they mention that if your manifold is semi-positive then the usual Gromov-Witten invariant can be built without the finer notion of stable maps.)
Think of it this way: You could also take the one-point compactification, but that probably won't tell you a lot.

Convergence to cusp curves is the original compactification by Gromov, whereas convergence to stable maps is the compactification by Kontsevich (a cusp curve corresponds to the image of a stable map). The latter is more accurate if you want to correctly model the topology of the moduli space, so as to build ``virtual (pseudo)cycles'' and use the full power of the Gromov-Witten invariants (this being in the realm of algebraic topology). I believe an explanation is in Chapters 5+7 of McDuff-Salamon's big book. (There they mention that if your manifold is semi-positive then the usual Gromov-Witten invariant can be built without the finer notion of stable maps.)
Think of it this way: You could also take the one-point compactification, but that probably won't tell you a lot.