I can say that error 3003 is the most annoying error I got in Diablo 3. As I read on the forums, Blizzard is working hard to fix the login problems most of us have since Diablo 3 was released. Some issues had been fixed already and I'm very curious how the login will work smoothly tonight. Beside the errors from the launch day, today I saw on the forums a lot of people complaining about the error 33 and error 3003. After I did a little research I found that we get error 33 at login in Diablo 3 when the battle.net servers are down for maintenance. The same error appeared with hours before the launch, when the servers weren't live yet. So, if you get error 33 when you login in Diablo 3, all you can do is wait, because it means that the servers are down to apply a patch or fix.
On the other hand, the new errors 3003 and 3006, that many Diablo 3 players get at login, are the main concerns to the Blizzard staff, and they are working very hard to resolve those issues. Blizzard said: While we've identified the causes and are working to resolve them, we expect our efforts to require an approximate two hours to complete. It is obvious that this is caused by the fact that millions of players are trying to connect all at once, and keep in mind that every retail account had guest passes, so the actual number of player trying to login, is larger that the sales.
Anyway, I hope Blizzard will fix very soon the new issues called Error 33, Error 3003 or 3006. The game is way to epic to be affected by this kind of technical issues.