"Vengeance" of God is a symbol of God's "Punishment" of the nations by the severe consequences of their rejection, violation, perversion of God's established Laws, Commandments.

"Vengeance of the Lord" is a spiritual battle of the righteous, people who know and fulfill God's Commandments with people of "pagan" worldview and vicious way of life.

Negative Meaning.

"Revenge" of man is a symbol of explicit, open, fierce or hidden confrontation of sinners, wicked people to the Righteous, their spiritual and educational activities, their exposure of sinful acts, false beliefs, exposing pernicious ideologies, pagan traditions of both secular and religious leaders and their followers.  

"Revenge" of a person is also the infliction of cruel offense, infliction of pain, open or hidden manifestation of anger, hatred of another person to satisfy his ego, pride.

"Revenge" has a variety of motives, is manifested by a variety of methods and forms both overtly and covertly.