Paulwp as others have said, the voltage is the same on both pairs of wires but the currents are not. To get a grasp of this think of a tiny electric motor and a really big motor hooked to a battery. Because of the impedance difference in the motors they will draw different currents from the battery. If two resistors were hooked to a battery the amount of current flowing thru the resistors would depend on their values.
In a tweeter if it was connected directly to the amp, it would try to move with the bass. It would probably quickly burn out. The crossover (the simplest is just a cap in series) acts as a high impedance at low frequencies and because little current flows in a high impedance circuit no force is generated on the cone of the tweeter at low frequncies.
In the woofer the crossover (just a coil in the simplest) as the frequency gets lower the coil passes more voltage to the woofer which causes more current to flow thru it. In other words at really low frequencies the woofer 'feels' all the voltage that the amp puts out.
To sum this up, at the crossover input terminals both high pass and low pass sections see the full voltage of the amp. At the output of the crossover (the driver terminals) only part of the full amp voltage is felt which is totally dependant upon the frequency. Since the voltages applied to the drivers terminals change with frequency so does the current. It is this change in current with frequency in the high/low pass circuits (which include the wires) that I was describing.
It should be easy to rationalize that the big currents flowing thru the woofer could not possiably flow also in the tweeter circuit or it would surely burn it out. If the speaker is connected with only a single pair of wires then both woofer & tweeter currents are combined at the amp but get seperated out in the crossover. In a bi-wire speaker the currents leave the amp together but get sepersted out at the amp terminal and only the tiny high frequency current flows into the wire connected to the tweeter crossover.
Hope this helps ... I will be out for 5 days and therefore can not respond unless I get on the net in Az. Chris