Different types/constructions of power cords DO make a "difference". Some cords increase the amplifiers mid/upper bass output a tiny bit, and some reduce it. Sometimes, rarely, there is an effect in the RF region that can poison the rest of the system. Such changes can have a big subjective effect and is a viable way of tweaking a system lacking real EQ or setup abilities.
But, don't go crazy, one can have pretty much any type of response below $300 on power cords; don't bother with 4 gauge $2k cords though. You can either loosen the coupling to the house wiring (smaller, higher inductance cords) or tighten the coupling to the house wireing (larger, lower inductance cords), but you are never going to get any "tighter" than what the 14gauge wiring in the wall will allow.
My testing was done with spectral sweeps and pink noise tests with Bryston amps on a fairly average, not too hard to drive, speaker. All the differences between cords centered around the power supply frequency. I should note that the flattest response into my speaker was with the stock cord. On a really nasty woofer load, the results might have been different. It might be fun to find a really nasty speaker load and redo the tests!