Rock climbing, broadly speaking, is the act of ascending steep rock formations. Normally, climbers use gear and safety equipment specifically designed for the purpose. Strength, endurance, and mental control are required to cope with tough, dangerous physical challenges, and knowledge of climbing techniques and the use of essential pieces of gear and equipment are crucial. Although rock climbing is an outdoor activity, many cities are home to indoor rock climbing gyms which can be formatted to match the physical (but not necessarily mental or technical) skill level needed for outdoor climbing (Wikipedia).
Mountaineering is the sport, hobby or profession of walking, hiking, trekking and climbing up mountains. It is also sometimes known as alpinism, particularly in Europe. While it began as an all-out attempt to reach the highest point of unclimbed mountains, it has branched into specializations addressing different aspects of mountains and may now be said to consist of three aspects: rock-craft, snow-craft and skiing, depending on whether the route chosen is over rock, snow or ice. All require great athletic and technical ability, and experience is also a very important part of the matter (Wikipedia).