The picture tube stores the charge. It acts as a large capacitor.
A capacitor consists of two conductors, separated by an insulator. In a simple example, picture yourself, shuffling on carpet. You are "conductive" enough to be one plate, the light switch cover is the other conductor - the air between you is an insulator. Get close enough, and the charge can bleed off you and into the switch cover, without making physical contact.
The outside of a picture tube is coated on the back in a conductive paint, called "aquadag". This outer coating is grounded to the metal frame of the monitor via a spring or strap across the back. The inside of the tube is also coated with a conductive coating - an this inner element is connected to the anode cup - that scary suction cup looking thing. Notice how the coating on the outside doesn't go all the way to the hole. These two coatings are the conductors - the glass of the tube forms the insulator. In operation, the flyback generates high voltage (about 20 thousand volts), which is fed to the anode cap. The frame of the tube (and outer coating) is grounded. The picture tube's inherent capacitor is part of the circuit, filtering the high frequency HV.
When you shut the monitor off, it's possible for a stored charge to exist in the "capacitor" formed by the picture tube. By discharging it, you're connecting a grounded screwdriver, and shorting out the capacitor formed by the tube. If you hear no zap, that means that either there was never any HV to charge the tube (I do believe, your case), or some monitors have a resistor to bleed off the excess HV after powerdown. An odd phenomenon is that the tube sometimes "recharges" itself from sitting. It's nowhere near as potent as the full charge, but touch that anode hole and you could get a startling zap. I've been bitten by that a few times - usually when carrying disconnected tubes. It's enough to startle you, but not painful or harmful.
A fully charged tube is enough to cause some minor injury, and hurt like hell, but it would be very, very unlikely to kill you. I think you're more likely to injure yourself by jerking your hand back into the frame than the voltage itself. Touch it while it's on and running, and it'll probably knock you on your butt. The voltage is very high - but the current is very low. Not lethal. Just painful.
-Ian
