Its main benefit is that is sounds cool.
And that it lets you access any file from anywhere that has an internet connection
That isn't really unique to cloud storage. For example, I can make the same claim if I email something to myself using webmail. The primary implications of cloud storage are that it is theoretically less secure, and it is theoretically more stable. For example, if you email something to your hotmail account, if hotmail goes down you can't access your data. If somebody blows up the building that has the server storing your data, it's forever gone. Whereas if you upload to a cloud since redundant copies are kept on a variety of different physical machines, if any one of them is lost or inaccessible for some reason, all the other redundant copies still remain. Which also results in the reduced security, since there are multiple machines storing your data. Though that depends on the nature of the particular cloud implementation. For example, if you have 1000k of data, with 100k stored on each of ten different servers using unique encryption methods in ten different countries each with different laws regrading data seizure, anyone attempting to retrieve a full copy of the data would have to go to a great deal more effort to acquire a full copy.
I would say that distributed data storage is appropriate for some purposes and maybe not ideal for others. But yes, like Dutchling says, it's mostly popular because it "sounds cool."