I was asked this question today. Here is some of my response: One is the emotion, the other is the action. Wrath is a human term for attempting to describe God’s feelings and inner response to evil. But God doesn’t sin in his anger like we do, so the topic of anger in God is difficult to describe. “the anger of man does not produce the righteousness of God.” (James 1:20). We often can’t control our anger. God is perfectly patient and merciful and loving and kind even while He has anger. Whatever way God may experience emotion is much different than we do; so much so that many scholars don’t even think it can even be considered to be in the same category. God is therefore sometimes described as being “impassable” or unable to be affected by emotion. Whatever the case, it must be said that if God has an emotion of anger, it is only because He has allowed such an emotion to occur. In other words, nothing happens to God unless he has already seen it coming and permitted it. In addition, we also usually only experience 1 or 2 emotions at a time. Since God sees and knows everything all the time, so He is simultaneously full of joy while also having this godly, holy anger toward sin, all of the time, every hour. So God’s wrath is not the same as his justice but they are definitely linked. God’s justice is the result of his wrath. I think a good example of that is found in Zephaniah, where it describes what a day of God’s wrath is like, which is really God doing a good thing, that is, punishing evil. In other words, doing justice:
A day of wrath is that day,
a day of distress and anguish,
a day of ruin and devastation,
a day of darkness and gloom,
a day of clouds and thick darkness, Zephaniah 1:15.
Justice is what all of us deserve every day. We are only surprised by this kind of wrath and justice because it is uncharacteristic of God. Usually He is gracious and merciful toward sin and rebellion, so we are caught off guard when we see humans getting what they deserve. Even the above description of wrath from Zephaniah contains some mercy because humans have not been brought to an immediate and complete and utter end.
To conclude, here are some quotes from the Lexham Theological Wordbook I found helpful:
“The wrath of God or divine wrath are concepts used in both the OT and NT that invite readers to consider the anger of God in contrast to his mercy. The concept of divine wrath emphasizes the danger of opposing the divine will and expresses, in human terms, the emotional reaction provoked in God by sin and rebellion. . . . It is important to note that the idea of wrath or anger is a human emotion that the biblical authors apply to God in an attempt to explain the consequences of turning from or disobeying God . . . the biblical notion of divine wrath is closely associated with God’s holiness, judgment, and compassion (e.g., Num 14:18). (Jim West, “Divine Wrath,” ed. Douglas Mangum et al., Lexham Theological Wordbook, Lexham Bible Reference Series (Bellingham, WA: Lexham Press, 2014).
Thankfully, for those who have turned to trust in Christ, “...God has not destined us for wrath, but to obtain salvation through our Lord Jesus Christ, who died for us so that whether we are awake or asleep we might live with him.” 1 Thessalonians 5:9-10.
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