You're absolutely right, in both instances You're climbing things (even the same mountain occasionally) it's just the means by which you get there. The TL;DR explanation is "climbing" is specifically centered around vertical cliffs, while mountaineering is a more all-encompasing sport focused on sumitting mountains. Basically climbing is more specific than mountaineering.

If you're curious to know more, look into the Yosemite Decimal System, it is how climbs/mountains are rated in difficulty (in the US) and gives a quick numerical value as to how hard a mountain is to summit (via a particular route) and gives a rough idea of what to expect.