These pieces of advice do not contradict each other because each is wise in its own way. Proverbs 26:4 means that in replying to a fool one should not descend to his level by giving him a foolish response (e.g, 2 Kings 18:36). Proverbs 26:5 means that one should correct a fool so he will not conclude that he is right (e.g, Nehemiah 6:8; Job 2:9-10). Some of a fool"s comments do not deserve a reply ( Proverbs 26:4), but others require one ( Proverbs 26:5). In unimportant matters one should ignore the foolish comment, but in important matters one needs to respond lest others conclude that the fool is correct. [1][source]
"In other words, it depends on the fool, and the truly wise person will be so sensitive to human nature that he will know when to apply the one and not the other." [2][source]