What's the difference between the two or are they the same? are there any other similar words in terms of meaning? In the context where you're pleading someone, which is better?
"Let me off" has a informal meaning of "don't punish me".
(student to teacher) Please sir, let me off. I won't do it again. You let Hopkins off with just a telling off when you found him smoking behind the bike sheds.
It is also used for "allow to disembark". "He stopped the bus to let the lady off."
"Let me go" means "release me". So it is used when someone is physically captured, whether for punishment or not:
After checking the bird for disease we let the bird go and it flew up into the tree.
So the expressions have a different meaning.