To weep bitterly is to cry, showing strong emotion related to pain and despair.

To be bitter is to be angry, hurt, or resentful because of one's bad experiences or a sense of unjust treatment. Imagine if someone wronged you and yet you KNEW you were in the right. Everyone sided with the other person and everyone may have judged you for your perceived wrongness. You have angry/confused/upset/vengeful feelings and think about it all the time. After a while, those feelings turn inward you become bitter.

Amargamente