The Rucksack Club does all of these, so could be called a "mountaineering club", but we equally welcome walkers who have no mountaineering experience, so we stick with the name "Rucksack Club" (which the club has had since it began in 1923!).
Mountaineering on snow-covered peaks in winter involves some extra skills and knowledge, such as using crampons and an ice axe, awareness of avalanche risk, and on harder routes, ice climbing. At the start of each winter season the club organises a special trip to give members the opportunity to learn and practice these skills. The club also arranges lectures on subjects such as winter walking skills, avalanche awareness, and navigation. There are also subsidised training courses available through the club.
Scotland's mountains are big by UK standards, but relatively small by global standards, the highest (Ben Nevis) being 1344 metres above sea level at its summit. Higher mountain ranges such as the Alps in Europe have many summits over 3000 metres (Mont Blanc is 4810m). Travelling abroad to climb higher mountains means knowing how to cope with less oxygen in the air (altitude acclimatisation) and encountering glaciers (in Scotland the winter snow melts away every spring/summer so we don't have any glaciers here). Walking across glaciers safely requires special techniques to minimise the danger posed by hidden crevasses and knowledge of how to rescue someone if they have fallen into one. Glaciers also continually erode the surrounding mountainside, causing rockfalls, unstable ground, steep slopes and narrow ridges, so an awareness of these dangers and how to deal with them is also important.
The best way to learn and develop the necessary skills is to take a mountaineering course in the Alps. There are several subsidised courses available for students and young people each year, and many members of the Rucksack Club have benefitted from these. Many have then gone on to organise their own Alpine Mountaineering expeditions using the skills they learned.
Ski Mountaineering involves special ski equipment including ski bindings with releasable heels, "skins" to stick to the base of the skis to prevent them slipping backward when you ski uphill, special boots which can also be used for walking, and avalanche safety equipment.
As Ski Mountaineering requires a certain level of skiing ability and a lot of specialist equipment, it has a more limited appeal, so the Rucksack Club doesn't formally organise Ski Mountaineering activities. That said, there are usually a few members around with an interest in it, so if you are keen there is the opportunity to meet other like-minded folks through the club.