Cables certainly can make a difference: if they’re single-ended (3-wire) they create the high crosstalk problem that the “balanced”/bridged headphone amps and cables (4-wire) were designed to solve. 
Now will the difference be audible? Hard to say, as there’s no definitive data on the threshold of crosstalk audibility, but it’s more probable you will hear it the lower your headphone driver impedance is (so planars and 16-ohm IEMs are most likely to suffer from this) and the higher the common GND wire’s resistance is in the SE cable you’re upgrading from (like if the L and R return wires join together very soon after exiting the drivers, somewhere inside one of the cups let’s say, and then there’s a single GND wire running the whole length of the cable to the amp - this is one SE design that worsens the common ground resistance, and thus the crosstalk).
So with 16-ohm headphones as an example, we have:
Common ground path: 0.5 ohm (wire) + 0.1 ohm (connector)
L or R signal path: 16 ohm (driver) + 0.5 ohm (wire) + 0.1 ohm (connector) + 1 ohm (amp output, let’s say)
The crosstalk due to the voltage divider formed where the return wires join into a single GND is:
20*lg( V_common / V_total ) = 20*lg( R_common / R_total ) = 20*lg( 0.6 / (0.6+17.6) ) = -29 dB.
This might be enough to not hear stereo performance problems according to some people, while others like NwAvGuy said -60 dB was the ideal to strive for (but I’m seriously doubting this second number more and more as I look at the numbers, because only 600-ohm headphones - the most immune to this - can achieve this kind of separation, while everything lower than that we’re supposed to believe is audibly subpar for separation qualities like width and imaging, if running SE).