In a summary of those links, imagine that FTL transportation is easier than FTL communication, and that there are no causality violations 'cause reasons. So, rather than really long wires running from star to star, you have analogs to the Pony Express and Carrier Pigeons. Voluntary runners (really, every ship that can spare the space) are automatically paid for having a box on their ship which handles all the storage, uploading, and downloading, and each planet has a "post office" system to handle routing and "layovers". Messages get from star to star reliably and cheaply, but slowly. Think days to weeks, like snail mail.
Note: this doesn't become the only network; the regular internet works just fine, but only on a planetary scale 'cause light speed.
What I'm looking for in this question is how people would use the internet if it worked like this. Personally, I think it would be a lot less like "browsing" and a lot more like "inter-library loan". Usenet, StackOverflow, and maybe modified versions of Facebook and Reddit would work, but IRC, Twitter, and real-time gaming would probably be limited to individual planets.
What do you think? What would the novel and/or utilitarian implications of the system be if this existed? What would its use look like to the end user? How would we go about doing all the myriad things we use the internet for today, but in space? How would this interact with each individual planetary internet?