Purple suggests it's 5v VSB but it could possibly be used for something else.
It would make sense however for a 5V VSB line to power the LCD, this means the LCD receives power and operates even when the machine is powered off but still plugged into the wall. - Only if it's hooked up with another purple wire from the PSU e.g. off the ATX 20/24 connector. or... if you hook it up to a USB/PS2 header powered off 5V VSB, e.g. when your optical mouse is still lit red even when you shut down.
If this is the case, you would still be able to hook up the red wire to the LCD but leave the purple wire out.
But, the colours may be non standard so there is a risk of damaging the LCD.
I'm out of ideas now, so... Good luck
EDIT: One more suggestion!
If you're like me, make mistakes all the time and bought the case off eBuyer;
Give it a try anyway, I once fried a VFD by sending 5V up the wrong wire, sent it back to eBuyer for a new one
Ok, that was a bit naughty, but to be fair the case (the VFD came with) and the VFD connections weren't labeled, just like yours!