For one, forsake is much stronger in meaning than leave because there is the additional element of finality to it.
if you forsake somebody , you leave someone forever/abandon him/her, especially when they need you most
Moreover, forsake is predominantly used in literature/the Bible or formal speech
eg
My God, my God, why hast thou forsaken me?
That you would be faithful, not deny Him, forsake sin, live or die for Him, or witness to your neighbor?
hope this helps.:-D