In biology, the term non-native does NOT = invasive
A non-native plant will become a problem if it becomes invasive.
A plant is called invasive if ecological and/or biological controls are not in place to help keep said plant in check. Many non-native plants become invasive because we usually don't import the controls when we import the plants. If said plant also has a high rate of reproduction and finds an efficient means of dispersal, it will most likely become invasive.
A non-native plant that becomes invasive becomes a real problem when eradication attempts are difficult due to rate of spread and/or difficulty of extraction (such as Japanese Ligustrum and Bermuda Grass along riparian zones in central Texas) AND when their growth begins to replace native plants and even alter ecosystems (such as Chinese Tallow on the Gulf Coast).
A native plant can BECOME invasive if native ecological/biological controls are removed or inhibited by humans or natural disasters. Such plants are usually called pioneer plants and will invade a disturbed habitat until conditions stabilize and/or alter to accommodate succession (such as Aspen Trees in the Rocky Mountains).