Good Ending: Kratos finally forgives himself (or his family does), and he no longer is bound to repeat the cycle of death and revenge because of his inability to let go of the past. The ashes on his skins falls off, the chains that binds the tools of death to his body slink to his feet, and Kratos finally embraces his family and departs from his personal war.
Bad Ending: Unable to forgive himself (and who could?), Kratos has killed and destroyed EVERYTHING, and yet, even as he stands upon the top of Mount Olympus, even if he has been covered with the blood of all the gods, even if he is free of fate, even if he become the victor of the final war, he still cannot let go of the past, and nothing has changed.
Giving up, he either continues to be the God of War, referencing the first ending, or he plunges himself off Mount Olympus.