It lends itself well to adventures that require abilities other than out-right monster bashing, since PCs are so well equipped with skills.
And due to the plethero of races, and the relationships between them, there is a richness to Conan characters that makes intrigue plots very attractive.
With very little useful magic, mystery plots are much easier. Players have to get out and ook for clues instead of casting a few divination spells.
In a standard fantasy world, Orcs vs humans is usually a clearcut choice. In a purely human conflict, which side to join is not usually so clear-cut. One side might be wearing guard uniforms, but that doesn't make them the good guys - their opponents might be bandits, but they might equally be heroic freedom fighters rising up against an unjust lord.
Also, given the innate mystery involved with the supernatural, it lends itself very well to games in which fear of the unknown monster is just as important as battling the monster itself.
This is harder to do in D&D, since players will generally try to deduce what the creature is, and when they have done so (or think they have) they lose all fear of it.
I'm planning an "alien"-esque scenario in which the players hunt and are hunted by a solitary, fast moving, stealthy critter in an underground maze. In D&D, they would soon say (eg) "Oh, its just a choker". In Conan, they will be ignorant and therefore fearful right until the end...
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Before somebody says it, yes, you can run all those sort of plots in D&D. In truth, I usually do, disdaining high-level, high magic monster fests. So I won't be changing my style, but Conan plays to that style much better than most rulesets (except WFRP, maybe).