The poem opens by telling how the speaker got angry at his friend. He told his friend about his anger, which got it out of his system. His wrath at his friend ended. However, he does not tell of his anger to his foe or enemy.

I think he does not tell his foe for two reasons. First, he might not trust him enough to reveal that he felt hurt or injured. He might believe that instead of sympathizing and apologizing, his enemy would laugh at him or repeat the injury. Second, since he apparently already sees this person as a foe or opponent, he might be looking for an excuse to be angry or offended at him. If you don't like a person, your tendency is to try to find rationales for your dislike. The speaker may have wanted to nurse his sense of injury rather than get over it so that he wouldn't feel guilty about his hatred.