Leh or Ulaanbaatar?

Neither city has any swimming to speak of. Leh's snowmelt rivers are for rafting and looking at, whereas in Ulaanbaatar local families at least paddle in the shallow Tuul in high summer, and a hotel pool is the only real place to swim.

Both kitchens are hearty rather than refined, but Ulaanbaatar gives you more to choose from. Besides buuz, khuushuur and stone-cooked khorkhog, a capital's range of Korean barbecue, Russian food and steak houses runs all year, along with a proper beer and vodka culture.

Leh's momos, thukpa and skyu are warming. The rooftop cafés along Fort Road and in Changspa serve trekkers well, and the Indian menus make it the easier of the two for vegetarians. But much of it closes outside summer, and the apricots are the most distinctive thing on the table.

Both cities have equal cultural weight, and both draw on a Buddhist past, but you see it in different ways. In Leh it is the townscape itself: the nine-storey palace, the Namgyal Tsemo ridge and a mud-brick old town of chortens and lanes, with only modest museums.

Ulaanbaatar is the place for museums. The gilded bronzes in the Zanabazar collection, the Bogd Khan's Winter Palace and the Chinggis Khaan museum stand among Soviet blocks, with Gandan's monks chanting close by. Each takes two or three days at most.

Ulaanbaatar has a real night out, small as it is. Around Sukhbaatar Square and Seoul Street there are pubs, karaoke rooms and a few large clubs playing local hip-hop until late at weekends. Leh has none of this: a beer with a rooftop dinner is the whole evening, and the cold and the dawn starts send everyone to bed early.

Leh's surroundings are on a different scale. Treks through Hemis National Park start within an hour or two, the road climbs to Khardung La for the mountain-bike descent, rafts run the Indus and Zanskar gorges, and Pangong Tso and the Nubra dunes lie further out. Winter adds snow leopard tracking and the frozen Chadar trek.

Ulaanbaatar's nature is gentler and closer to the ground. You can hike the larch forests of Bogd Khan Uul from the southern suburbs, ride across the Terelj valleys or look for takhi wild horses at Hustai. In return there is no altitude to adjust to, though city traffic eats time on the way out.

Here Leh is far ahead. Thiksey and Shey are less than half an hour away, Hemis about an hour, and the thousand-year-old murals at Alchi make one of the finest days of early Buddhist art anywhere. The outings are nearly all monasteries, but they are of the highest quality.

From Ulaanbaatar the usual choices are the steel Chinggis Khaan statue, the ruins of Manzushir and a visit to a herding family. The great historic site, Kharkhorin with Erdene Zuu, needs an overnight stay.

Ulaanbaatar is the more workable choice with children, mainly because of what lies outside it: ger nights, horse rides and marmot-spotting, plus the Gobi fossils at the dinosaur museum. The city itself, with its traffic and broken pavements, is less friendly.

Leh brings families to high altitude in a single flight, so young children risk altitude sickness, and the outings mean long drives on mountain roads. It works only for older teenagers who are keen on rafting or trekking.

Both centres can be walked, though in Leh the climb from the Main Bazaar to the old town is hard work in the thin air. Ulaanbaatar is a little easier because ride-hailing and cheap buses fill the gaps, even if the traffic is heavy. Leh depends on the fixed-rate taxi union, hired motorbikes and shared jeeps. In both cities, any trip into the countryside means hiring a driver.

Ulaanbaatar has the better shopping, and cashmere is the reason: factory outlets and the long-established State Department Store on Peace Avenue sell good sweaters and blankets. There are also felt crafts and horse-head fiddles, and Narantuul market is rough but fun.

Leh's Main Bazaar has pashmina, thangkas, turquoise jewellery and solid trekking gear. However, the same stock repeats from stall to stall and quality varies, so you need to check carefully what you buy.

Leh is the cheaper place to stay, with family guesthouses and plates of momos costing very little. The extra costs are the taxi union's fares and the permits for the outer valleys.

Ulaanbaatar is cheap at street level, but central hotels and Western-style restaurants cost closer to Southern European levels. The driver, guide and vehicle you need for the steppe can cost more than everything else on the trip.

Leh has no theatre or concert hall, and its performances are religious ones. Masked cham dances fill the monastery courtyards at the Hemis festival in summer and around Losar, the Tibetan New Year. Seeing one in its original setting is deeply moving if your dates match, but on other days there is little to go to beyond the odd cultural evening for tourists.

Ulaanbaatar is far better for a night at a show. On most summer evenings its traditional performances put the same masked dance, here called tsam, on stage alongside khoomei throat singing, the morin khuur fiddle, long song and contortionists. The salmon-pink opera and ballet house on Sukhbaatar Square adds European classics and Mongolian works, and in July Naadam brings ceremony and music to the city.

Neither city is built for pampering, but Leh gives more room to slow down. You get quiet guesthouse roofs, meditation courses at the Buddhist centres and consultations with amchi healers, although the thin air disturbs sleep for the first few days.

Ulaanbaatar's noise, traffic and smoky winters make it tiring, and it offers only hotel spas and Korean saunas. Real rest comes from leaving for a ger camp in Terelj.