Everything that users were complaining about being broken on launch day

Load/Saving, regional commerce, great works, talking with friends, leaderboards, trading (including implicit trading like commuters), regional unlocks from city halls, etc. Plus cheat detection.
There's probably some other things I'm missing.
I personally don't remember anywhere we claimed that your local city wasn't simulating on your computer. Doing it otherwise would be insane. Even fully synchronous online games like League of Legends do a significant amount of gameplay simulation on your local machine to help hide latency, and part of our scalability and multiplayer design was that online play would be generally asynchronous--you could do stuff in your city while I'm not online and vice versa.
EDIT: It's awesome how having a 'maxis' logo on your name in a detailed, open answer leads to downvotes. Stay classy, /r/SimCity. Please don't turn into /r/gaming.