Gods are immortal and have particular domains that They control/embody/personify. Demi-gods, or heros, are descended from god and mortal pairings. Demi-gods have abilities that surpass mortals, can cross domains that gods can not cross (like descending into Hades – something we don’t see any gods but the Chthonic gods do, because it isn’t Their realm [except for Hermes and Iris because They are messenger gods]). Yet mortals don’t have the power of gods and can be subject to the whims of gods. Mortals often have a special relationship with their parent god and sometimes are blessed with long life or immortality by their parent god or gods in general. Sometimes they have/had cult worship as well (as in, people pray[ed] to them in addition to the gods).
IMHO, Catholocism adapted the practice of worshiping demi-gods/heros to the worship of their saints (not the same figures in mythology/history, but rather, the very human tendency to elevate certain notable individuals to cult status and ascribing a divine quality to them).
What do you think a demi god is? What do you think defines a god?