The main difference is that MobileMe allowed you to store and share ANY file you had. With iCloud (right now anyway) you can only store iWork app documents, it's automatic, and you can only share them with other Apple devices you own. Now there is the iWork.com beta that you can share those iWork documents though, but I suspect that it will be rolled into iCloud at some point.

At some point, other apps will be able to take advantage of iCloud storage, in fact the reminder app Remember The Milk already does as of this week.
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