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.

The way the story goes, the mountaineer that "discovered" the range was frustrated he couldn't summit the mountains, as he was a mountaineer not a climber, so he named them the unclimbables. Ironically enough, just over 5 years later it saw its first ascent. It's now considered one of the 50 best climbs of North America,

The main wall for the climbing community here is the Lotus Flower Tower, the flat looking one in the middle of the group of 3 on the left. Bucket list climb for sure.