What worked for me is what someone else said to do in this thread. Thought i'd reiterate in a bump for people just checking the last page:
1. Go to the Origin Application.
2. Click the "origin" dropdown menu.
3. Click "Application Settings..."
4. Click the "In Game" tab at the top
5. Uncheck the box marked "Enable Origin In Game"
6. Save, and close the program.
7. Run Origin as Administrator.
8. Instead of running Simcity through the Origin Client, Run as Administrator from Desktop Shortcut.
Win!! Running as we speak.
EDIT:
This fixed the problem of not running after clicking play, but when I entered the game, it loaded the tutorial until the very end ("tightening graphics") and then said "cannot connect to servers at this time". I think however they're working to fix this issue right now. Will respond with progress ASAP.
RE-EDIT:
Well the tutorial finally loaded. I actually got to the very end, right up until I had to go to Lucky Shores (or whatever it's called). It started to load the Lucky Shores region, but halfway through I got the error message "Could not connect to server at this time". So at this point, I don't think it's necessarily anything with the program itself, rather than hugely problematic server issues, which EA themselves have acknowledged. If you have the same problem as the OP, this fix really should work. If you have trouble claiming cities or loading, it's on EA's side. My guess is that the loading/claminig issues will be fixed by the end of the day. Good luck everyone! Hope EA at least learns from this whole experience.