Non-indigenous (non-native) species: with respect to a particular ecosystem, any species that is not found in that ecosystem. Species introduced or spread from one region of the US to another outside their normal range are non-indigenous, as are species introduced from other continents.
Invasive species: a species whose presence in the environment causes economic or environmental harm or harm to human health. Native species or non-native species may show invasive traits, although this is rare for native species and relatively common for non-native species. EO 13112 actually uses the term invasive species to mean invasive non-native species, as does this paper.