People make distinctions between different activities in order to sort them into concepts that they can readily understand. To me, backpacking and mountaineering are incredibly similar, but there is a distinction that is hard to identify. As a backpacker, peak-bagger, climber, it is those special trips that combines aspects of all three to make a mountaineer. Perhaps I see mountaineering as backpacking with a precice purpose, including a technical challenge. Sure hiking the AT is extremely challenging, but it is not mountaineering. Yes, climbing El Capitan in Yosemite is technically challenging, but misses the some of the remoteness required for a true mountaineering ascent.
For me, mountaineering tasks involve a certain amount of risk, some technical aspects, and some challenge. Climbing the gullies to obtain Mount Elbert was mountaineering, but hiking the regular route seemed like a very long hike. Climbing Longs peak by the keyhole route felt like mountaineering, because the last section required some scrambling at (relatively) high altitude, though the hike did not require an ice axe or a rope. I invite you to come up with your own definition of mountaineering. As you can see, I am an emerging mountaineer, gnawing at the bit for more.