They both transmit heat very quickly. It's true that copper transmits heat faster than an equal thickness of steel, but the stainless tubing is thinner walled so you can't actually make that comparison.
Whichever is faster, though, doesn't matter in practice: both exchange heat fast enough that it's your water flow rate (as opposed to any minimal insulation from the tube walls) that governs the heat transfer.
Some people think there's a benefit to including copper in the pre-fermentation brew system. Stainless is cheaper and easier to clean. Opinions vary on which is prettier.
Use whichever you like.
On deck: Little Bo Pils, Bretta Off Dead (Brett pale)
Secondary: Oude Bruin, Red Sky at Morning (Sour brown ale)
On tap: Saison Duphunk (sour), Amarillo Slim (IPA), Earl White (ginger/bergamot wit)
Bottled: Number 8 (Belgian Strong Dark Ale), Eternale (Barleywine), Ancho Villa (Ancho/pasilla/chocolate/cinnamon RIS), Oak smoked porter (1/2 maple bourbon oaked, 1/2 apple brandy oaked)