Tokyo or London?

Neither is a beach city, but Tokyo has the better swim within reach. The Shonan coast is about an hour south by train, with long beaches of dark volcanic sand at Kamakura and Enoshima. In August the water is warm enough for a comfortable swim, though the bathing season is short and weekends are crowded.

London's nearest sea is the pebble shore at Brighton, where the Channel stays cold even in late summer. Its real swimming is in fresh water: the lidos at Brockwell Park and Parliament Hill, and the Hampstead Heath ponds, where some people swim all year.

Both are superb places to eat, but Tokyo goes deeper. Its strength is specialisation: counters that have made one dish for decades, from station soba to sushi and kaiseki. Eating well there also costs less than visitors expect, and the depachika food halls and even the convenience stores are worth exploring.

London's strength is range. There is Bangladeshi cooking on Brick Lane, Punjabi food in Southall, Turkish grills on Green Lanes, Borough Market and the pub Sunday roast. It is excellent, but meals cost more, and most of its best cooking comes from its immigrant communities rather than from a home tradition refined to Tokyo's level.

London has more to see. The British Museum, the National Gallery, the V&A and the two Tates each fill a day, and most are free. The landmarks run without a break from the Tower and Westminster Abbey to Wren's St Paul's and the Georgian squares of Bloomsbury.

Tokyo lost most of its older buildings in the 1923 earthquake and the 1945 bombing, so its heritage survives in pockets: Senso-ji in Asakusa, Meiji Jingu, the Imperial Palace moats and the lanes of Yanaka. The museums of Ueno Park, led by the Tokyo National Museum, are a real draw, and the modern architecture is worth seeing in itself, but there is less to fill a long week.

Both give a first-rate night out, but the evenings take a different shape. London's is built on the pub and on districts with their own character: Soho for late cocktail bars and its gay scene, Shoreditch and Dalston for warehouse parties, Brixton and Peckham for big venues. Many pubs empty around eleven, so the weekend is the time to go big.

Tokyo is known for the very small bar, a few seats built around one owner's passion. They are packed by the hundred into Golden Gai or stacked floor upon floor, with izakaya around every station. The last train near midnight sets the pace: go home early, or stay out in Shibuya or Roppongi until the first trains in the morning.

Tokyo's trains reach real mountains, and London's do not. Mount Takao is an easy forest hike about an hour away. The Okutama valleys, inside Tokyo's own boundaries, have river gorges and steeper ridges, and Hakone and the Fuji Five Lakes give the classic views of Mount Fuji, which can be climbed in summer.

London has more green space within the city itself, with Hampstead Heath, Richmond Park and its deer, and Kew. The countryside beyond is gentle, though: the Chilterns and the South Downs offer good walks through beechwoods and over chalk hills, but nothing dramatic.

Both are excellent bases with fast trains in every direction, and London's list runs a little longer. Its outings are mostly university towns, cathedrals and old English history: Oxford and Cambridge about an hour away, Windsor Castle, the Roman baths at Bath, Stonehenge with Salisbury, and Canterbury.

Tokyo's day trips lean towards temples, shrines and hot springs: the Great Buddha and Zen temples of Kamakura, the Toshogu shrine at Nikko in its cedar forest, the onsen inns of Hakone, and the Edo-period streets of Kawagoe and Sawara. Kyoto, though, is too far for an easy day.

Tokyo is the easier city with children, less for its attractions than for how simple everyday life is there. The streets are safe and clean, stations have lifts and baby rooms, spotless public toilets are everywhere, and the food suits fussy eaters. Disneyland and DisneySea, the Ghibli Museum and Miraikan fill the days.

London has a strong offer of its own, much of it free, from the dinosaurs of the Natural History Museum to the pirate-ship playground in Kensington Gardens, with Legoland Windsor outside the city. The hard parts are the stairs at many Tube stations and the crowds when schools are out.

Neither needs a car, but Tokyo's trains are in a different class. It has the densest rail network in the world, trains run to the minute, signs are in English, and nearly every sight is a short walk from a station. The weak points are the maze of Shinjuku station and the network stopping around midnight.

London's Tube, the Elizabeth line and its buses reach everywhere, and the top deck of a double-decker or a Thames riverboat works as sightseeing too. But the city is spread out, many stations mean long corridors and stairs, and travel between outer neighbourhoods takes time.

These are two of the world's great shopping cities, and Tokyo edges ahead on specialist shops. London's strengths are tradition and character: bespoke tailoring on Savile Row and Jermyn Street, historic department stores such as Liberty and Fortnum & Mason, and weekend markets from the Portobello Road antiques to the Columbia Road flowers.

Tokyo goes deeper into particular trades. There are anime and electronics in Akihabara, kitchen knives on Kappabashi, books in Jimbocho, vintage clothing in Shimokitazawa and Harajuku, and fine lacquer, ceramics and washi paper. Even a small purchase is wrapped with great care.

Tokyo is clearly the better choice for money, though a room is not cheap in either city. In Tokyo the saving is in eating: ramen, set lunches and conveyor-belt sushi are cheap and good, trains are fairly priced, and many shrines and gardens are free, so most of the bill is the hotel.

London is expensive across the board, from small central rooms to pub rounds and theatre seats. The free national museums and parks are a great relief, and the markets and South Asian or Turkish cafés help, but it never feels inexpensive.

London is the stronger city for the stage, and for an English speaker the gap is wider still. The West End playhouses, the National Theatre, Shakespeare's Globe and the fringe stages above pubs give drama huge range. The Royal Opera House, five major orchestras, the Wigmore Hall and the summer Proms at the Royal Albert Hall cover the rest.

Tokyo's offer is deep but harder for a visitor to follow. Kabuki at the Kabuki-za, sold one act at a time, Noh and the Takarazuka Revue are things London cannot offer. Suntory Hall and the small live-music venues of Shimokitazawa and Koenji are excellent, but spoken theatre is almost all in Japanese.

Tokyo is much the better place to unwind, because bathing is part of daily life there. Neighbourhood sento, some fed by natural hot-spring water, big complexes with saunas and outdoor pools, and the onsen inns of Hakone are all easy to reach, though many baths turn away guests with tattoos.

London's options are a handful of old Turkish baths, the hotel spas and a cold dip in the Hampstead ponds. The pace there is fast, and for proper bathing you go to Bath.