I can only echo Seyorni's wonderful explanation and Mystic'Heather's valiant effort to shed understanding on the matter.
My take on karma is much more immediate. Karma is Newton's law, "For every action the is an equal and opposite reaction" but with a twist. Karma, from my standpoint is "for every action there is an equal reaction of the same nature". In effect, so-called "good" actions will result in favorable reactions from the world around you. So-called "bad" actions will yield unfortunate results. It is the knowledge of this simple process that one takes with them from life to life as they wend their way to "enlightenment", not the karma itself. In effect, life deals with a person's actions during a given lifetime and you either learn from the experience or you do not. The individual has direct influence on their personal karma and their personal karma is NOT dependant on the external influence of some imaginary "invisible friend". NO ONE, but the individual, can absolve you of your karma, period. Karma is an autocorrecting mechanism that enforces responsibility for actions.
Sin, on the other hand, is more like a cheque book, in my opinion. The individual can indeed learn to "sin no more" but they still retain their accumulated "sins" and have to be hopeful that an external agency (aka "invisible friend" or "gawd") will absolve that imagined sin. It is both filled with hope and also with hopelessness, as we are told we are all inherently sinful creatures and that only "god" can remove our sins entirely. It's a bit much to hope for, in my opinion.
Since karma is up to the individual, and sin is up to god, they cannot be the same thing. Perhaps they have similar effects in life, in that the individual theoretically become a "better" person, but they have decidedly different cleansing methods. In effect, karma uses life to autocorrect behavior; whereas sin relies on a rather large stick of fear and the carrot of reward as the impetus. To me, that is unseemly and unrealistic. Give me reality any day.
Oh well, what would I know, eh?