I have Anno 2070 which is one of the Ubisoft games that uses the new system. If it cannot find the servers then you cannot access your previous saved-data and the software is coded to disable certain functions in the menu. Also, whilst the original campaign (a very short 6-mission thing) is available with the game, you cannot play any other missions (called World Missions iirc) without being logged into the Ubisoft servers, and if you lose connection whilst playing the game, then a load of things get disabled mid-game, such as being able to store goods in your main 'base' because that info is stored online.
Edit : As I understand it, losing connection in Diablo 3 is even more catastrophic because of the procedural generation, it means that, whilst the achievements your player has may be preserved, the game you were playing will have actually changed by the time you get back in.