Shawshank easily, not to say that the Green Mile was bad but Shawshank was just so good.
Oddly enough they both were directed by the same guy, Frank Darabont.
It's not to say the Green Mile wasn't a good movie, just very few movies can match-up against Shawshank and win. Very few.
“The most terrifying fact about the universe is not that it is hostile, but that it is indifferent. If we can come to terms with this indifference and accept the challenges of life within the boundaries of death, our existence as a species can have genuine meaning and fulfillment. However vast the darkness, we must supply our own light.” - Stanley Kubrick
Shawshank for sure. That film was strong in so many ways, and, it had an absolutely great performance from Morgan Freeman, who's narration in that film is almost hypnotic. The only part of that movie that I feel drags at all, is the arrival of the kid, who Tim Robbins helps get his diploma.
Green Mile is good, but it drags a bit at points. The performances are good, but Hanks has done much better, and Michael Clark Duncan's performance was good, but not spectacular, I found.
Also, I'm suprised you didn't mention that both films are directed by Frank Darabont.
I give the edge to Shawshank although both are in my top ten. I think although you can draw similarities in many ways (period peices, prison dramas, Stephen King stories, Frank Darabont directed etc.) that the two movies in themselves are much different. Shawshank was about philosophy and how one lived their life, whereas the Green Mile sent home a message about the force of life as a whole. I suppose perhaps the best way that you can show the huge difference between the two in my opinion would almost be the cell-blocks they take place in. Shawshank was a massive prison for a movie about grand ideas and concepts, whereas The Green Mile took place in a small almost peaceful row, which reflected a movie which was much more personal.
Anyways, just some thoughts I felt I'd throw out there. As I said earlier, Shawshank takes it for me, though by a closer margin than most people would give.