I'm a few months late but I can add some more details on the differences (in case anyone needs to make up their mind).
The two versions are *mostly* identical in terms of content, so it will not matter a huge amount which one you choose.
Uru: Complete Chronicles has (compared to Myst Online):
- Single-player only.
- Works offline, no DRM.
- A more coherent story (especially the start of the game, which guides you into the game world as Cyan originally intended, instead of just dumping you into it with no explanation).
- Several puzzles and areas that are not present in Myst Online. In particular, the D'ni areas have been repurposed from "multiplayer gathering" areas into story-driven puzzles. And the "Path of the Shell" plot line has an actual ending as opposed to Myst Online where it is just a collection of Ages.
- A few puzzles changed so as not to require a friend to complete.
- Several other puzzle changes.
- A journal feature allowing you to write notes down in-game.
- As Renyaloth mentioned, lack of widescreen support.
Myst Online: Uru Live has (compared to Uru: Complete Chronicles):
- Persistent online connection required.
- Completely free to play (and no in-game payments).
- Focus on multiplayer. Although almost all of the game can be played solo, there are a number of puzzles that require a second player's help. There are also a number of areas and optional activities that are designed solely for interacting with other players. Keep in mind that the online playerbase is very small these days, so you may not be able to find anybody online to help you. Your best bet is to play with a real life friend.
- A handful of Ages and areas that are not present in the single player version. All of these are small compared to the single player Ages, but they probably represent a few more hours of gameplay.
- More modern engine, with widescreen support.
In summary, both versions have content not present in the other version (there is unfortunately no "complete" version of Uru, despite the name). I would suggest that whether you have a friend to play with is the most important factor: if you have one or more friends to play with, go with Myst Online. Otherwise, stick to Uru: Complete Chronicles; you will have a smoother and more coherent experience. And if you really enjoy it, you can always play the remaining content in the other version (mostly without having to replay the rest of the game).