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It's mainly for the screen, but I find it okay for the case as well.
It sounds like you've actually scratched the case, though. I'm pretty sure the aluminium is painted, and so, it may not feel deep enough to be a scratch, but it's probably that the paint has been scratched off. If the marks look like a dark shade of grey, or black, then this is probably true.
Real dirt should just lift off with the likes of cleaning wipes.
Good luck,
onlyone-jc.