Let me explain it from my point of view. I own and operate a one or two man software company that also hosts web sites. I work in the flim & tv music industry, meaning I have a shit load of music (literally terabytes) that has to be available for download.
8 years ago I owned a rack of servers downtown here that I managed myself. Honestly, it wasn't that bad. I bought reliable used 1U servers (mainly IBM and Dell) off ebay and stocked them with disks. I ran FreeBSD and Linux, used RAID, etc. But I always had two issues to deal with. The main one was "I have to always be available to handle hardware issues".
My company isn't big enough to hire someone to do it, but I managed for nearly 10 years with no disasters. In that time I had a motherboard crap (when I was starting out with one server - ouch) and a few disks fail. In all of those times I had to go in - sometimes in the middle of the night - and fix/replace whatever was wrong.
Then I found Amazon AWS. Here's the kicker - it was actually cheaper for me to simply "rent" storage from them than to rent rack space for my own servers. I moved my servers to linode.com - again it was cheaper although they're nowhere near as fast as my former dedicated servers were, but they're fast enough for my applications and I can always move to larger instances where needed. And that eliminated my maintenance issues for hardware while costing less per month and maintaining the same 3-4 nines level of availability that I've always had. Oh, one other thing - S3 makes it just as easy to secure my audio files but the delivery speed can easily saturate any pipe that the files are being delivered to.
So the cloud might not be "magical" and solve all the world's problems, but for small IT shops it's great. Everything I do is on the internet so the whole "what if your connection goes down?" issue doesn't exist for me. I do not recommend such a solution for everybody. I have clients in the industrial wholesale space and their inventory & sales system definitely should be on-site with off-site backups. But their web site can be hosted elsewhere.
Anyway, yes, the "cloud" is very useful for many businesses.