They most definitely do, everything in the signal path and attached to a transducer like a cartridge makes a difference, a big difference in some cases.
Materials and construction improve as one spends more on headshells, wires and mounting hardware. This translates to better detail retrieval and micro-resolution. Naturally, the higher the resolution of the system, the more apparent these differences will become.
I think that if you are spending decent money on a cartridge, you ought to put it on a really decent headshell, with premium wires and have it aligned correctly and meticulously. I’ve found headshell wiring to be especially important and currently use SME silver headshell wires. They are the best I’ve used, out of many sets of Litz, 99.9999.% pure copper, etc wires. Silver wire is generally a game changer wherever you use it, but it is very expensive.
Other wires work well too, like Ortofon’s silver wires, Jelco’s Litz wires and some Audio Technica wires. I supply basic sets of wires which work very well, through to high-end wires, whatever you might like to use with your deck.
Headshells, fasteners and even the rubber gaskets at the headshell/tonearm interface are important and all of them contribute to the final result. I always suggest using the very best headshell you can afford and making sure that it matches the mass of the tonearm and the range of cart/headshell weight that your arm can support.
Naturally, what ties this all together is the right advice, so getting this is critical. All too often, I see and hear about the nonsense that leads to terrible-sounding setups, so I am on a mission to move away from that. Many turntables are hobbled by poor cartridge-tonearm matching, poor headshell, wires and fasteners, and of course poor setup and calibration.