ESD-safe tweezers have a black coating on the handle. If I am grounded to a work bench, and normal tweezers are already conductive, then what is the point of the conductive black coating? Screw drivers sometimes have handles which may be insulators, but tweezers are all metal. Are they not as conductive as I am expecting? Maybe an oxide layer?
The tweezers that you are describing have what's known as a Static Dissipative coating. This coating is conductive, but it has a very high resistance.
What this means is that they are able to dissipate any differential charge slowly/safely, without there being a very low impedance path to ground that might produce an arc.
Put simply, the tweezers will not prevent static discharge, but they will limit the peak current at which this occurs, which can be enough to save some (but not all) components from destruction.