In the Christian culture, you ask for forgiveness and you receive it. Admitting fault and saying "sorry" puts the wrong in the past. The Christian god just wants his followers to ask forgiveness, and he gives it. And Christians are told to grant forgiveness to other people, in much the same way.
Heathens have a set process for correcting a wrong, and gaining what Christians would call "forgiveness." Our ancestors believed that if you do something wrong, you should fix what you broke. You should pay for the wrong you've done. And when you've fixed the wrong or paid for it, then (and only then) the matter is put behind you.
So, if a heathen tells a friend that he will help him do something important, but something comes up, and the heathen fails to help his friend....he doesn't just say "sorry." He should go to the friend and say, "I know I was supposed to help you, but I didn't. That was wrong." And then the heathen should offer to help the friend with something even more important and follow through. Or perhaps the heathen would give the friend something that would make up for letting the friend down.
If a heathen breaks something that belongs to someone, he replaces what was broken or pays for it. If a heathen unintentionally says something that causes unintended problems for someone else, he does what he can to correct the problem or make up for it. If a heathen says he'll do something or be somewhere, and he fails to follow through, he goes to the person he made the failed commitment to and compensates them in some way for the failure.
When correcting or paying for a wrong you have done, sometimes it is sufficient to decide how to fix it or make up for the wrong and just do it. But often, especially in important matters, it is better to go to the person who was wronged, and ask them exactly how you can fix it or how you can make up for it. If the person wronged names a reasonable action or payment they wish you to make, then you accept the Shyld (payment or obligation) that they set and follow through on it.
When a wrong is done, the intentions of the wrong-doer are not of paramount concern. The primary focus is on the actual wrong...the actual injury that was done. Whether that wrong was intentional or accidental, it was still wrong-doing and still needs to be fixed or payment made. The injured or wronged party may look at the intentions of the wrong-doer when setting Shyld, or when considering whether the wrong-doer will repeat the injury in the future. But accidental wrong-doing still demands the wrong be corrected or payment made.
In very small matters between friends or kindred members, things that really aren't a big deal, we give each other a "pass" sometimes. A simple acknowledgement of the small wrong that was done and a statement of regret will usually suffice in small matters. But, even in these small matters, it strengthens a friendship when you actually care enough to take steps to make up for a small wrong you've done.
The problem with the concept that a simple "sorry" fixes all wrongs, is that in practice it does not actually work. The wrong is not actually addressed, and even if the wronged person accepts the apology, it is just human nature to continue to be upset about the wrong because it has not been fixed or paid for. Only by actually addressing the wrong and showing through one's deeds that you recognize your own accountability for what you did, is the matter actually repaired and properly set aside.
In not giving forgiveness easily, it is not that heathens are mean, or pointlessly vengeful, or hold endless grudges. They just expect that when someone wrongs them or hurts them, that they should make up for it in some way. It restores the balance. The reciprocity in the relationship. And honor is restored.
Honor is really at the heart of this Heathen process of correcting a wrong you have done. This process restores the honor of both parties involved. Among our ancestors, the wronged party would take vengeance upon a wrong-doer in order to restore their honor and the honor of their family. And a wrong-doer, in correcting or paying for a wrong, shows his own sense of honor in trying to fix what they have broken.
Mark Ludwig Stinson
Jotun's Bane Kindred
Temple of Our Heathen Gods