The concept of "local copy" is not going to disappear anywhere. For files that are needed to be at hand reliably, local copy on hard disk drive or data crystal or whatever media is used is never going to be overtaken by remote storage. Same with local processing vs. remote processing... But, I can imagine remote storage and processing being useful in some cases - in fact, some things already use a setup like this.
Consider Opera Mini, a relatively popular phone browser software. What it actually does is that it connects to Opera's server, sends the url, which is then processed by Opera's server hardware into a form that is sent to phone, which then shows the data. It's advantages are that the phone doesn't need to do as much page processing, so the loading times are reduced (in theory anyway). Disadvantages are that it adds a third party to the connection so if you want secure connection, it might not be appealing to you.
Also, remote processing might provide an advantage for cases where the local processing capacity is limited...
But wait, is any of this actually new technology? SSH connection has existed for a long time already. The only thing that limits the usability of remote console and desktop solutions is transfer speed. Using X11 tunneling with SSH, I can already do this stuff on my university account; I can store data there and use any software installed on the university computers that I have access to. The reason why I don't do it is that with current internet connection, the GUI gets tacky. Data storage has come off useful on occasion - having a public_html folder is pretty handy for some limited file/image hosting.
But, as said, for confidential, private data and cases where reliability is needed, local copy will always be preferred solution. Remote processing... I can't think of applications where more processing power would be needed for normal users. Current PC's are not going to get any slower either, and professional/scientific simulation runs will be done on dedicated environments anyway.
So I kinda fail to see the point. All they are offering is a way for users to store casual, meaningless crap on their servers, which I can't understand to be honest.
Then again, if it proves reliable and secure (I have my doubts seeing how it's Microsoft, and Google is starting to give me shivers anyway...), it could be utilized as an SVN storage space for projects such as FSUpgrade or any mods/tc's with large enough staff for it to be meaningful.