It's depends on the "kind" of mountaineering. There's a large amount of "mountaineering" that doesn't involve "climbing on rock or ice" in the strictest sense of class 5 routes needing trad gear or ice pro, but still involves technical roped glacier travel. There's also a large amount of "climbing" which doesn't involve "mountains" (or even the outdoors for that matter).
I think what makes the difference between "hiking or scrambling" and "mountaineering" in most people's minds, though, is whether the route is at least in some way "technical". In this sense, rock climbers often have an advantage because they understand technical rope systems and protection, are often more comfortable with exposure, etc. All a climber has to do to be "mountaineering" is be climbing a route to the summit of a mountain. Conversely, when a hike becomes a mountaineering objective can be a bit fuzzier. I've heard some people say it's "big glaciated peaks" that makes it mountaineering. I've heard some say it's any time any sort of "protection" comes out, even if that's just an ice axe. So some would say scrambling isn't mountaineering because there's no protection involved . But ironically, the American Alpine Club doesn't recognize scrambling as distinct from climbing at all, and in their yearly Journal of Climbing Accidents they list the "secondary cause" for all scrambling-related accidents as "Free-Soloing" since they just see it as a form of unroped climbing (Anton Chigura: "which it is").
There's also a healthy pipeline from backcountry skier/split-boarder to mountaineer, FWIW.
Mountaineering tends to be very outdoor interdisciplinary. In that way, it's kind of the /out/ endgame.
