New York or Tokyo?

New York has the better swimming during a city stay. The subway runs to Coney Island and the Rockaways, with a boardwalk, surfers and open Atlantic sand, and the Long Island beaches are about an hour out. Tokyo's own bay shore is port and reclaimed land, so a real swim means an hour on the train to the dark volcanic sand of Kamakura and Enoshima.

Sagami Bay is a touch warmer in August, but in both cities the season is short and summer weekends are packed. Neither city is worth choosing for the beach.

These are two of the world's great eating cities, and they feed you in different ways. New York's strength is range. Whole cuisines are cooked by their own communities in Flushing, Jackson Heights and on Arthur Avenue, alongside the city's own slices, bagels and deli pastrami.

Tokyo goes deeper rather than wider, with counters that have perfected one dish over decades, the depachika food halls and the yakitori alleys at Yurakucho. It edges ahead because eating very well there costs much less, whereas in New York the top end is where the bill hurts.

Both cities have museums for a long stay. The difference is in what surrounds them. New York's Met, MoMA and Museum Mile sit among landmarks the world knows by sight, and the history you see is the brownstones, cast iron and tenements of the last two centuries.

Tokyo lost most of its older city in 1923 and 1945, so its heritage is scattered: Senso-ji, Meiji Jingu, the palace moats and the formal gardens, with Ueno's museums holding the great collection of Japanese art. Go to New York for Western art and skyline icons, and to Tokyo for temples, gardens and an artistic tradition you will not see at home.

New York has the bigger night out and no curfew. The subway runs all night, so a cocktail bar on the Lower East Side can turn into a Bushwick warehouse party at dawn without any planning. Its range, from West Village wine bars to gay bars dating back to Stonewall, is hard to match.

Tokyo's night is smaller in scale and more personal, built on the tiny six-seat bars of Golden Gai and the izakaya around every station, with clubs in Shibuya and Roppongi. The last train around midnight forces a choice between going home and staying out until morning, but the streets feel safe whichever you choose.

Tokyo reaches real mountains more easily. Mount Takao is an hour away by train, the Okutama gorges lie within Tokyo's own boundary, and Hakone and the Fuji Five Lakes put the view of Fuji within a day out.

New York's outdoors is mostly its designed parks and waterfronts, such as Central Park, the High Line and the birdlife of Jamaica Bay. Breakneck Ridge in the Hudson Highlands is the best proper hike you can reach by train. Both cities are at their finest in the autumn colours, October in New York and November in Tokyo.

From New York the standout is Philadelphia's colonial old town. Princeton, New Haven, the Hudson estates and Dia Beacon are quieter alternatives, but Boston and Washington really need an overnight stay.

Tokyo makes the stronger base. Kamakura's Great Buddha and Zen temples, Nikko's shrines in cedar forest, Hakone's hot springs and Kawagoe's old merchant street are all an easy train ride away, and they show a side of Japan the capital itself has largely lost.

Both cities have the big draws: New York has its dinosaur halls, the Bronx Zoo and the free Staten Island Ferry, and Tokyo has its Disney parks, the Ghibli Museum and Miraikan. The real gap is in everyday life.

In Tokyo the stations have lifts and baby rooms, the toilets are spotless and the streets are safe and clean, so a day with a pushchair goes smoothly. In New York, stair-only subway stations, small hotel rooms and the Midtown crowds tire everyone out, and staying near a park makes a big difference.

New York does well. Manhattan's grid is easy to walk and the subway runs all night, but the mix of local and express lines, weekend reroutes, stairs and the trip to and from the airports all take effort.

Tokyo's rail network is denser and runs to the minute, with English signs and one card for everything, and almost every sight is a short walk from a station. Its weak point is the reverse of New York's: the trains stop around midnight, leaving only taxis.

Fifth and Madison Avenues and grand old department stores such as Bergdorf Goodman and Macy's give New York the classic big-city shopping day. SoHo's cast-iron streets have the fashion brands, Williamsburg and Greenpoint have vintage, records and young designers, and the specialist side runs to rare books, sneakers and weekend makers' markets.

Tokyo is level on luxury, with Ginza's houses and the century-old Mitsukoshi, and on vintage in Harajuku and Shimokitazawa. Where it goes further is craft and whole specialist districts: kitchen knives on Kappabashi, books in Jimbocho, lacquer, ceramics and washi paper, and stationery stores worth a visit for their own sake, with every purchase wrapped with great care.

Tokyo is the much cheaper trip, mainly because eating out is a bargain. Ramen, set lunches and conveyor-belt sushi are good and cheap, and many shrines and gardens are free. Rooms are the main cost, and they rise sharply in cherry-blossom season.

New York is expensive at almost every turn, from small Manhattan rooms to tax and tips on every meal and high museum fees. Pizza slices, the restaurants of Queens, the free ferry and the parks help, and staying in Brooklyn or Queens brings the cost of rooms down.

New York is the stronger stage city, and much easier for visitors to enjoy. Broadway's musicals and plays are in English, Lincoln Center has the Met Opera, the Philharmonic and City Ballet, and Carnegie Hall, Village jazz clubs and Harlem gospel fill the rest of the week.

Tokyo's stage life runs deep too, with kabuki at the Kabuki-za, Noh, the Takarazuka Revue, concert halls known for their acoustics and countless small live houses. Most of its spoken theatre, however, is in Japanese.

Tokyo is the more restful city. Bathing is an everyday habit there: hundreds of neighbourhood sento survive, some using natural hot-spring water, alongside large bathing complexes, and Hakone's onsen inns are about ninety minutes away. Temple grounds and quiet backstreets add calm, though tattoos can still keep you out of some baths.

New York has a bathing scene of its own, with the old Russian and Turkish baths in the East Village, banyas in Brighton Beach and Korean spas. Outside them, though, the noise and pace of the city work against rest.