Hope you all know that "rime" also means "snow".
Mariner is frantically longing for an outlet. He wants to share his experience with someone which makes him feel relieved. It is like a catharsis or unburdening. This can also be read as the creative travail of every writer or artist.Every artist undergoes a period of extreme stress and tension (almost like the labour pain of a woman)before he is successful in giving vent to his ideas. Mariner longs for creative output.
Mariner's experience or journey is not one man's journey alone. It represents the entire journey of humanity. True to the Bible, every one is a sinner. Or, as per other religions every sinner has to face nemesis. It need not always be a punishment from outside. Our mind carries every experience as memory and it is enough to punish us. As T.S. Eliot says in "The Waste Land",
"Keep that dog far hence which is friend to man"
He refers to our conscience as a dog which would unearth all the sins we have buried in the subconscious of our mind. Read Dostoevsky's "Crime and Punishment".
The punishment for Mariner is to wander cautioning people against the kind of sin he has committed. He is also like Jacob Marley in "A Christmas Carol" who is doomed to wander in chains. Mariner says he has killed the Albatross for no reason. Yes, Sin is the result of ignorance. The bird he had killed dominates his psyche. That's why he compares the storm to a bird in his narration.
The hardships undergoes is a symbolic representation of the consequences of sin/crime. In crime/sin one may be with a group. But in punishment one is alone. No fellow sailor comes to his support. They,in fact, blame him for killng the bird. They punish him hanging the dead albatross round his neck.
This poem also has hidden reference to the crucifixion of Christ who like the Albatross came to save or help humanity. The same cross we carry round our neck like the albatross.
Read this poem and "A Christmas Carol" side by side. You will get more insights.