2 Corinthians 7:8-11 illustrates that repentance is not mere regret or sorrow. It does this by using, and directly contrasting, three terms: "sorrow" (Greek noun lupē, verb lupeō), "regret" (here used in forms of the verb metamelomai) and "repentance" (metanoia). Indeed, the point is made that the sorrow or regret "of the world," leads to death:
8 For though I made you sorry (elupēsa) with my letter, I do not regret (metamelomai) it, though I did regret (metemelomēn) it. For I see that my letter made you sorry (elupēsen), though just for a while. 9 I now rejoice, not that you were made sorry (elupēthēte), but that you were made sorry (elupēthēte) to repentance (metanoian). For you were made sorry (elupēthēte) in a godly way, that you might suffer loss by us in nothing. 10 For godly sorrow theon lupē produces repentance (metanoian) to salvation, which brings no regret (ametamelēton). But the sorrow of the world (kosmou lupē produces death. 11 For behold, this same thing, that you were made sorry in a godly way (theon lupethēnai), what earnest care it worked in you. Yes, what defense, indignation, fear, longing, zeal, and vengeance! In everything you demonstrated yourselves to be pure in the matter.2 Corinthians 7:8-11 (WEB) (parentheticals added).
Note carefully what this passage asserts about the relationship between regret, sorrow and true repentance. First, note that repentance does not arise from mere regret. If I do what I know is wrong, but excuse myself by saying that "regrettably, it had to be done," this is not repentance. Knowledge of the pain my actions cause others, or God, and regretting that pain, is not repentance. Likewise, remorse occasioned by being caught and punished--regretting the consequences of my actions that fall on myself--is not true repentance.
On the other hand, true repentance may arise from sorrow about my actions. Specifically, repentance arises from "godly sorrow"--that is, sorrow which comes from God, for His always-restorative purpose--not from the sorrow of this world. It should be remembered that the background of this passage is the rebukes Paul delivered to the Corinthian church in his first letter to them. In his earlier letter, Paul had reproved them for dividing into parties that honored human leaders and followed the wisdom of the world (I Corinthians 1:1-3:23), for tolerating sexual immorality, including a member who had his father's wife (1 Corinthians 5 & 6:12-20), for wronging each other and suing each other to redress those wrongs in pagan civil law courts (1 Cor. 6:1-11), for dishonoring marriage by improperly depriving each other of marital rights and through divorce (1 Corinthians 7), for injuring each others' conscience by eating meat sacrificed to idols (1 Corinthians 8 & 10), by eating the Lord's Supper self-centeredly, not dealing with their divisions, and thus not recognizing the Lord's Body (the Church) in partaking of it (1 Corinthians 11:17-33), for not maintaining proper order in worship and abusing the gifts of the Holy Spirit in the process (1 Corinthians 12 & 14), and above all for pursuing showy demonstrations of God's presence and power rather than love for each other (1 Corinthians 13). Paul's First Letter to the Corinthians was, in many respects, quite harsh, and it did cause sorrow in Corinth.
However, though Paul initially regretted reproving the Corinthians so harshly and causing them grief--though he did so under the inspiration of the Holy Spirit--by the time he wrote his second letter he no longer regretted his harshness. God had used the first letter to create "godly sorrow" which led to "repentance." But, as Paul explained, that "repentance" was not limited to a feeling of regret or to merely saying "I'm sorry." No, the repentance that resulted from godly sorrow worked "earnest care" in the Corinthians to correct the problems Paul had identified in his first letter. It produced "defense, indignation, fer, longing, zeal, and vengeance." The actions taken by the Corinthian church in response to the godly sorrow prompted by Paul's first letter were sufficient to demonstrate them to be "pure" in the matter. This is true repentance, and it always comes from God.
By contrast, the sorrow that comes from the world leads to death. Worldly sorrow does not come from God. In the first place, it is often sorrow over the wrong things--not sorrow that I did wrong, but sorrow over the consequences to myself and others. God offers forgiveness for sins. There is no forgiveness for consequences--only God's grace to overcome them, if I am willing to receive that grace. But if I am only "sorry" for the consequences, I am in no position to receive that grace. I must either continue in denial, or despair. Secondly, the sorrow of the world is often imposed on us by others for their own ulterior purposes. People use other people's ability to feel guilt to manipulate them into doing what the purveyor of the guilt wants them to do. I can all too easily be conned into a "guilt trip" about the real or imagined consequences to someone else of one of my actions--even an action that was good and right--and convinced that I must do something to compensate them for those consequences, that I owe them something. But these compensatory actions do not come from God, they come from the will of another human being playing on my guilt. They, too lead to death, and have nothing to do with true repentance.
True repentance, then, arises from sorrow that comes from God, and leads me back to God, back to earnest care to do what God wants.