Ζω (not ζώω ;) ) has the meaning of "I live, I am alive" and μένω means "I stay". So, αναπνέει άρα ζει=he breaths so he lives, but θα μείνω εδώ για λίγο=I will stay here for a while. But in this context above, both can be used. "I live in Greece"="Ζω/Μένω στην Ελλάδα".
Just wondering, this has nothing to do with this lesson but I see in a lot of places online Man is άνδρας, could that be alternate spelling?
And also είναι όμορφη vs Αυτή είναι όμορφη. they can both be used to say she is beautiful right? but the first one is more correct and the second can be used to call other things beautiful?