Hardware wise there is no difference beyond the minimum requirements of each MS OS, drivers not withstanding. WHS, 7, Vista, XP, Linux whatever all can share files over a network and can all be installed on the same brew of hardware. If you are just looking at simple file storage and perhaps a little RAID/file protection then I would suggest neither 7 nor WHS. A simple NAS or Linux RAID box (like FreeNAS or Openfiler) could be a nice option if you like to tinker even a little bit.
The big thing for WHS is the additional features it has over a file server/NAS. It is the best
platform Microsoft has put out imho. I would say 7 but MC really makes 7 what it is for me and Media Player drags it down so WHS wins. Beyond being a glorified file sharing/NAS solution WHS offers a ton of features. The usefulness of those features is up to each person though. If you live alone and only have one PC then the backup features are pretty useless. Also kind of makes centralized file storage a moot point if you only have one machine accessing it.
Check out MS's website for WHS. There is a nice features table somewhere in there. WHS is great for backup of PCs, sharing files, remote access, and a plethora of add-ins: http://www.microsoft.com/windows/products/winfamily/windowshomeserver/default.mspx