Intuition insists that there is no difference, but in practice, operating a field-reconfigurable dual-voltage motor at 120V/115V instead of 240V/230V will tend to reduce the available torque... but the discrepancy between theory (but it's the same!) and reality (no, not quite!) is caused by reasons that are entirely external to the motor.
The (correctly-wired) motor sees exactly the same conditions regardless of the series/parallel wiring of the motor leads -- as long as conditions are static and the load on the motor (the wood against the blade, its hardness, thickness, and the force pressing it into the blade) is not varying. Of course, with a table saw, operating conditions are quite variable.
A table saw motor draws more power when there's wood against the blade than when the blade is just spinning freely with no work to do. More work to do means increasing the current, which the motor does automatically as it "tries" to maintain is designed rotational speed under load.
Voltage drop in actual volts (not percentage) on a circuit of a given wire size and length is directly proportional to current draw.
Current drawn by a motor increases with workload.
Current is doubled when voltage is halved, so the voltage drop on the circuit is larger with the higher current required by the lower voltage. (The wire size is increased, of course, and this is another factor in voltage drop, but the reduced voltage drop from the larger wires is not sufficient to counteract the increased drop from the doubled current, unless wires much larger than code requires are used at the lower voltage.)
The current increase under load is also doubled when the voltage is halved.
The power supplier's transformer may be able to better handle the demand when the entire secondary is involved in supplying the peak current, rather than only half of it.
...so the motor typically experiences a more aggressive voltage drop under load when operated at the lower voltage.
And that's where things get ugly, because of another fact about AC motors: the rated torque is available for delivery to the workload only when the motor is operated at its nameplate voltage, and falls off following the law of inverse squares as supply voltage decreases. (Bad things also happen with overvoltage, but that's out of scope of this answer.)
As available supply voltage is reduced by increased voltage drop, torque decreases by a factor of the square the reduction from nameplate voltage... so a motor running at 10% undervoltage (90% of rated) develops torque of only 0.9 x 0.9 = 81% of rated capacity... and, as the physical load on the motor surges (e.g. when you slide a piece of wood into the blade) the current increases and the voltage drop increases more dramatically when configured for 120V than when configured for 240V.