No one can argue against that it's extremely well supported on the client-side and pushing files to a server (with ssh for example) or hosting a small http-server locally (twistd for example) is usually no big problem.
I've got a web server. Or I could host one locally. But it's all just so cumbersome and unwieldy for what should be a simple task. Take a file. Push it from Point A to Point B. But I'm technically competent enough to set up all that stuff and wrangle with it. Someone else who just wants to share some files, maybe not so much.
Realistically? I'm serving them via HTTP now when I use my Public folder. But the process for getting the file to the person I want to share it with has been grossly simplified.
Just because I am technically competent enough to go through a detailed process to share a file doesn't mean it's a good User Experience. I enjoy working on and maintaining this stuff, but it can still be frustrating. Dropbox makes it painless. Almost effortless.
I believe the "client" in this case was "customer". Getting somebody to right click on a folder and selecting 'share in dropbox' is the easiest thing that I can ask of somebody. All files within that folder will be visible to me, even if they make changes.
Another key thing with dropbox is that it made sharing files easier by allowing you to share folders. I, for one, going through the attach -> browse -> open dialogue for each photo I wanted to show to somebody.
If I had to change anything I was already doing to use Dropbox, I wouldn't have even given it a first thought, let alone a second thought, least of all my $10. And in your example it is the "client" that makes it so easy for the "customer" to get them to share files with you.