While forgiveness is important to some religious views, I do not think it is particularly important in this case.
When you excuse someone, you allow them to escape the consequences of their actions.
When you forgive someone, you cease resenting those actions.
One can forgive someone in your heart, while still holding that they must be punished or not relieved of misfortunes their actions brought upon them. One can excuse someone, but still resent them.
In the overlap, one is often taken to include the other, because the two do often happen together.
It's also more likely to speak of excusing an action that did not have any volition, such as an accident or eructation, and forgive of a deliberate action that led to some harm whether that harm could be foreseen or not. It would not be unheard of for the other to be used.
There isn't really any social rank matter, bar different etiquette rules as to what things one should ask to be excused for. In particular, saying excuse me after breaking wind or burping was once a classic "non-U" identifier, that is an identifier of someone who was middle class trying to pass for upper class (the working class at the time might ask it, and might not, the middle class almost always would, while among the upper class the polite thing was for nobody to pass any comment on it). Such class markers are not as firm as once they were.