Yes, the meaning of these two is very similar and I have added destiny as an accepted answer, but it is not exactly the same.
Fate: it's out of your control. Destiny: it's what you are meant to do. Destiny has slightly more positive connotations. You'd more usually say "he was fated to die in that accident" but "she was destined for greatness", but it is sometimes heard the other way around. Very similar words!
Are you discussing English or Greek? What I understand from Babiniotis μοίρα and πεπρωμένο are the same. Μοίρα is the μερίδιο (share, from this comes the geom. signif. degree) the 3 Mοίρες give to the newborn child. Nothing can change it, not even the gods.
Πεπρωμένο is from the ancient πέπρωται/ determined by the fate/ μοίρα. Also the saying "το πεπρωμένον φυγείν αδύνατον" / one cannot escape from the fate