Currently "mountain-climbing" is a master tag and "climbing" an associated synonym. I suggest it should be the other way round or even better two different (but linked) tags and ask you to discuss this proposal and ultimately mods to re-organize the tags.
Neither tag has a description (that I would like to add after the re-orga) but for me
- mountain climbing is about climbing mountains, ranging from expeditions to hiking as ways to scale a mountain
- climbing is more general, in my humble understanding it includes mountain climbing, rock-climbing or sports climbing (where you climb one length of a rope) in a gym or at an outdoor crag (that can be very different from a mountain), bouldering (again in- and outdoors, you have a mat and climb only low heights so that falling is still safe), ice-climbing (frozen waterfalls or giant freezers), Deep Water Solo (you climb without protection and fall into the water below), climbing trees, what else not ...
My suggestion is to make two separate tags "climbing" and "mountain-climbing" for now. At least one of these new tags will apply to all old questions. As the site grows eventually new tags such as "rock-climbing", "bouldering", ... might be added but for now I don't deem that necessary.