Yay for first post here!
I wouldn't break the bank, but I've learned over the last 5 years in my mastering studio that cables do play a part.
Analog variants will certainly be a little more revealing.
Digital cables are a little more finnicky to discern, but different protocols might make a difference IN ADDITION to the cabling.
For example when I go from to TOSLINK/SPDIF input on my Trinnov Nova to my RCA/SPDIF it feels like a veil is lifted. Not so much balance of frequency, soundstage, etc. But just a thin piece of fabric was removed. Had other people in the studio to listen as well. I can test this in a Single Click from my Nova Software. Both from my RME ADI-2 Pro FS R.
If I "brand swap" AES cables, really little to no difference IMO with same length. Haven't tested enough with SPDIF 75ohm. A good shielded properly rated operating impedance cable does fine enough. Lengths do absolutely matter too. There are other articles & studies on that.
However I've experienced with digital devices overall, the same cabling will reveal differences as well. IE My RME ADI-2 Pro FS R SPDIF out vs my Prism Lyra SPDIF out to my Trinnov Nova, yes I can strangely enough tell a significant difference. Did not expect that at all. FWIW they were using Belden 1694A with true 75ohm connectors. Prism was RCA to RCA with true 75ohm crimp connector. RME was DIY Crimp DB15 (or db9 can't remember) to Crimp 75ohm RCA. Highly doubt the DB15 vs the RCA connector will be the tale teller...
I try to avoid Optical because I feel there are better options of transmission IME.
HOWEVER back to the OP ask:
OP was asking about headphones, and I swapped out my junk 1/8 to 1/8 stock cable from my Slate VSX to Ghost Cable Analog and the differences were more discernable than above examples.
YMMV