Thing is, Conan as written is not the mindless brute that some depictions make him out to be. As put in the Wikipedia article...
Despite his brutish appearance, Conan uses his brain as well as his brawn. The Cimmerian is a talented fighter, but due to his travels abroad, he also has vast experience in other trades, especially the thiefly one; he is also a talented commander, tactician and strategist ... In addition, Conan speaks many languages, including advanced reading and writing abilities ... and his very first appearance (in The Phoenix on the Sword) shows him busy writing. Another noticeable trait is his sense of humour, largely absent in the comics and movies, but very much a part of Howard's original vision of the character...
I think that's probably what makes him appeal to me, aside from my identification with REH's own battles with depression. I live in a world where "make nice to bozos" is the most important job skill you can cultivate, but it's a skill that walks up and down my spine the way the sound of a dentist drill does. But Conan doesn't make nice to bozos, he gives them a well-deserved boot to the head.
(Okay, there is one story, "The God in the Bowl," in which Conan spends most of his time grunting and generally acting like a junkyard dog ... but in his defense, he was pretty hard-done-by in that story, so I suspect he was feeling a bit cranky.)
You'd think, with my fondness for wordplay and airy persiflage, that I'd be more inclined to obsess over Gray Mouser. I think the reason I'm more interested in Conan is that when I want to escape, I'm often feeling a bit cranky myself. ;) Mouser's hyperactivity can be more grating than soothing on those occasions.
On a vaguely-related note, jamesbarrett, you need to check this out.