I just bought Uru Complete Chronicles and having spent a while going through various wiki pages etc am very confused as to how this game does or does not relate to Myst Online or Uru Live.
As best I can tell, Complete Chronicles is just the single-player parts of Ages Beyond Myst, To D'ni, and The Path of the Shell, with the hooks to the multi-player parts taken out. Is that a good way to describe it?
If so, was it done completely? Or should I expect to find rooms, machines, references, clues, etc. that don't have any purpose now that they can't connect to the multi-player? (For instance, there are references to a KI in the documentation, but isn't that just a thing for communicating with other players? Do you still find and use a KI in the Complete Chronicles?)
Is there any way to download and play any of the "shards" that were released as part of Uru Online? Are those like extra single-player levels, or just muti-player stuff?
Is there any way to connect up this version of Uru with the recently rejuvinated Uru Online? Does that question even make sense?

You can use the KI to take pictures of anything in the game. Besides that, it serves some purpose within Ages Beyond Mist (ABM), but it's main use is during the To D'ni quest.
As far as I know, to D'ni was the planned multiplayer part of the original Ages Beyond Myst, but when Myst Online failed, they released it adapted to the existing single player part. So, whatever "useless clues" you would find in ABM are actually related to To D'ni.
I heard Shards are actually servers hosting Myst online Worlds, like any dedicaded server to a MMO.
I don't think that there's a way to connect URU: CC and Myst Online, but I wish there was a way to use at least my profile with everything I did already done.
I heard somewhere that Myst Online featured some new Ages, but I never found anything that would tell me what are these Ages and what is the new quest they bring.
You can get Myst Online on the official site: http://mystonline.com/
EDIT: you can get A LOT more information (and spoilers) about the Myst Series and the differences between URU:CC and Myst Online here: http://en.mystlore.com/wiki/Main_Page

These are some of the differences between Myst Online and URU:CC:
1. New Ages: Eder Delin, Eder Tsogal, Neghilan, Dereno, Payiferen, Tetsonot, Minkata. Jalak Dador.
2. Changed puzzles: you need people to help you in Eder Delin/Tsogal, Er'cana and Ahnonay.
3. The KI has been extended to include a journal and a communication system.
Other than that, it's almost the same game. Of course, new Ages should come soon to Myst Online.

Uru live is back up, and free.
Uru CC, (with Uru Live for free) is really only for the diehard Myst fan now. All of the ages are in Live, and they're free. In CC you can download custom ages with Drizzle though. I personally recommend Live.

Most of the new Ages have some sort of story behind them. Minkata for example was used as a training Age by the Guild of Surveyors, if I'm not mistaken. Jalak was used as a playground for an unknown game.
Most of the Ages in Uru Live are persistent. The events that lead to changes in the game can be read about online, in case you missed them, but the effects of these events can be experienced at a later time. However, at this point there is no official (e.g. Cyan) story in Uru Live and that is unlikely to change for the next few months. My prediction is that we will see some new stuff at the end of the year, maybe. As for expanding on the existing Uru storyline, well, maybe or maybe not.
tl;dr: I recommend playing Uru Live because it's Uru:CC with new stuff and with a great community of players. There are a lot of fun things to do so you won't get bored. However, buying Uru:CC is supporting Cyan since they are now a pretty small company.

What everyone else has said is pretty much true. Uru complete chronicles has the added bonus of a graphical upgrade and I think there's a few areas that are in that that aren't in the free Uru: Online, but those are small rooms, not entire ages (for example, going off of what I guide that I found said, there's a different ending to Path of the Shell in the origional version whereas the MOUL version just sort of "ends" after you get to the area that is revealed with the last Yeesha message. At least as far as I know; doesn't seem like there's anything to do in the last spot aside from collect a new cloathing item, grab a relto page and snag a shiny). Also, if screen shots are to be believed, there's the game in Gharaseen that's still playable in Complete chronicles that isn't playable in MOUL, although it's kinda pointless to not play that with a bunch of people.
MOUL has pretty much all of the things that Completle Chronicles has, though. New content is iffy; to my knowledge, Cyan has said that there won't be any new story but some things have been happening that suggest otherwise, so there might be more later on. We just can't expect it to be steady since it's a free thing now. Also, there's going to be an added functionality where players can create their own ages via Blender or 3ds Max. As far as I know it hasn't been intirgated but it's on its way.