The short answer is no. They are very similar but there are differences; some small, some quite significant.
To avoid spoiling, we will not detail specific differences here, just give an idea of the changes so people know what to expect.
The most significant point is that Myst Online: Uru Live (MO:UL/MO:ULa) puts you in a multiplayer environment, although you may not be aware of that at first, while the offline Uru is strictly single player. Many of the ages and puzzles are the same in both games so if you've solved those in one the same solutions will mostly work in the other. However some puzzles are altered in the online game to encourage co-operative play.
The first thing you may notice is that the location where you start the game is different: In MO:ULa, this is done to make it easier/quicker to access the "public" (multiplayer) areas, but can confuse people used to the way offline Uru starts.
Additionally, while the online game contains almost all the areas to explore that the offline Uru: Complete Chronicles* has (only a very few small areas are not accessible) there several additional ages that are not present in the offline version of Uru, and some will require you to work with others to complete those ages.
*If you've only played Uru: Ages Beyond Myst before, then Complete Chronicles also includes the "to D'ni" and "Path of the Shell" expansion packs.
Clothing options for your avatar are also slightly different.
Lastly, there is the prospect of new content being added to MO:ULa in due course (it may take a little while for new ages to be added, but some new fan-created clothing has already been added).
An additional note for new players: If you find yourself seemingly stuck at a scene with a sideways scrolling wall then don't panic - simply press the "Esc" key to move on (this is a temporary workround for another issue).
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