This is a known fault (see the Steam forums). The problem looks to currently lie with being a previously signed up EU member of PlanetSide 2, which Steam has a problem with. You keep getting redirected to their website and prompted to change your password, which eventually fails with a notice about needing to wait 24 hours.
The solution is simple, go to wherever Steam has installed PlanetSide 2, which for me was “C:\Program Files (x86)\Steam\SteamApps\common\PlanetSide 2″. Open up the “LaunchPad.ini” and add “environment=psg” at the end. My “LaunchPad.ini” file looks like:
The game will now allow you to log in with your username and password.