We analyze the relationship between two compactifications of the moduli space of maps from curves to a toric fano varieties: the Kontsevich moduli space of stable maps and the Ciocan-Fontanine--Kim moduli space of stable quasi-maps. We exhibit the moduli space of stable maps as a union of two (reducible) components: the moduli space of relevant maps and the moduli space of irrelevant maps. We equip these spaces with virtual classes such that their sum equals the virtual class of the moduli space of stable maps. The moduli space of relevant maps is birational to the space of qusi-maps and the enumerative invariants defined as virtual intersection numbers on the space of relevant maps are equal to the quasi-maps invariants. On the other hand, the virtual intersection numbers on the space of stable irrelevant maps are zero. This shows that Gromov--Witten invariants agree with quasi-map invariants for toric Fano varieties.
