The filesystem has gone away.
Everything about the previous generation of computers revolved around files. To a user, everything on a PC is a file. Nothing on my iPad looks like a file. There are icons for applications. Are they files? Or is this HTML presented in a browser? Does a user know or care?
This simple difference changes everything. No files means that DVDs, floppy discs, thumb drives, and all other forms of storage are suddenly repositioned. iTunes sucks in so many ways, but one way it doesn't suck is that I don't care where my songs are, just that there's a database of songs with an interface that is song-specific. Music playing applications are post-PC ways to manipulate music.
Obviously iTunes exists on a Mac/PC as well as on an iOS device. On a Mac or a PC, it provides an interface for dealing with music files, but I can still hunt for the files if I want to. If I move a file, I break the carefully crafted illusion of there just being music.
On a Mac, I install software by downloading it and dragging it into my Applications folder. I'm manipulating files. Sure, there are fronts put up so I can avoid the files, like installer packages and the new Mac App Store. But the application files are still there to be manipulated and I can still fool around with the Library and stumble over the filesystem's spoor when I go trekking through the Finder.
On an iOS device, there are no files, there is no Finder. Software is installed by some magic process that never exposes me to the implementation. The illusion is complete. This is what makes an iOS device a post-PC device. I imagine that there are or will be tablets and phones where I can break into the device and discover that hidden from the user is a world of files. Such devices won't be post-PC devices, they'll be PCs in a tablet form factor.
To summarize, my contention is that "Post-PC Device" doesn't mean tablet or phone, it means "No filesystem."