Mexico City or New York?

Neither city is a beach destination, but New York has the sea inside its limits. Coney Island, Brighton Beach and the Rockaways are at the end of subway lines, and Jones Beach is about an hour out, so a summer swim is easy to fit into a day.

Mexico City sits high on its plateau, hours from Acapulco, so swimming means a hotel pool. If sea and sand matter, plan a few days on the Pacific coast as a separate leg.

Mexico City rewards depth in one cuisine and its regions. Breakfast is tamales and atole from a cart, lunch is a counter seat at Medellín or La Merced among the chiles, moles and insects, and the night ends at a taco stand carving al pastor. Mezcal, pulque and the tasting menus of Roma and Polanco, built on regional ingredients, fill in the rest.

New York matches it with breadth. Almost every cuisine is cooked by the community it came from, Fujianese in Flushing, Tibetan in Jackson Heights, Italian on Arthur Avenue, and the city's own classics of the slice, the bagel and deli pastrami sit alongside them. Street and market food is cheap in both, though Mexico City's best cooking is spread across a very large city.

These two are evenly matched, but they show very different pasts. Mexico City layers Aztec Tenochtitlan, the Templo Mayor, a baroque colonial centre and Rivera's murals, with the Sun Stone in the Anthropology Museum at its heart and Frida Kahlo's house in Coyoacán.

New York's history is mostly from the last two centuries, seen in Brooklyn brownstones and the cast-iron fronts of SoHo. Its real weight is in art collections such as the Met, MoMA and the Guggenheim, along with the Statue of Liberty, Ellis Island and the skyline landmarks. Choose by whether ruins and murals or great galleries draw you more.

New York goes further and later. Trains run through the night, so you can move from an East Village dive to a West Village wine bar and on to a Bushwick warehouse that keeps dancing past dawn, with a different mood in every borough.

Mexico City has strong nights of its own. They centre on the mezcalerías and cocktail bars of Roma and Condesa, the old cantinas of the Centro and the salsa and cumbia halls where locals dance properly. The scene is more concentrated, though, and most nights end in a ride-hailing car rather than on a train.

Mexico City's outdoors is volcano country at altitude, while New York's is parks, waterfronts and the Hudson Valley by train. From Mexico City the pine forests of Desierto de los Leones and the slopes of Ajusco give real hiking, and the pass between Iztaccíhuatl and Popocatépetl or the crater lakes of Nevado de Toluca are about two hours out. Traffic eats into the day, though, and the air can be poor in the dry season.

New York's green space is easier to use. There are the woods and lakes of Central and Prospect Parks, the migrating birds of Jamaica Bay and kayaking on the Hudson in summer, and Metro-North reaches the rocky climbs of Breakneck Ridge in about ninety minutes. October colour is the best time for it.

Mexico City has the more striking outings. Teotihuacan's pyramids are about an hour away, and Puebla, Cholula, Taxco and Tepoztlán add tiled colonial streets, a silver town on a hillside and a giant pyramid with a church on top. Early starts are needed to beat the traffic.

New York's day trips are pleasant rather than unforgettable: Philadelphia's Independence Hall by train, the campuses of Princeton and Yale, and the Hudson Valley's Gilded Age estates, Dia Beacon and Storm King. Boston and Washington are too far to do well in a day.

New York has more made for children, and it is easier to reach. The dinosaur halls at the Natural History museum, the Bronx Zoo, the aircraft carrier museum and Central Park's playgrounds keep a family busy, and the free Staten Island Ferry rarely fails. Stair-only subway stations and small hotel rooms are the drawbacks with a pushchair.

Mexico City offers the zoo and the Papalote museum in Chapultepec, a trajinera on Xochimilco and lucha libre, and restaurants welcome children warmly. Long rides in traffic, uneven pavements and the altitude make it harder going with small ones.

Mexico City's Centro, Roma, Condesa, Polanco and Coyoacán are all good to walk, but the distances between them are long. The metro is big and cheap, yet its carriages are crushed at peak hours and a favourite of petty thieves, and heavy traffic slows the ride-hailing cars that most visitors end up relying on to cross town.

In New York the subway itself makes the difference. One network covers four of the five boroughs, so trains rather than cars carry you between almost every neighbourhood you are likely to visit, with ferries and Citi Bike to fill the gaps. Its quirks, such as express trains skipping stops, weekend diversions and stations with only stairs, slow you down now and then rather than costing half a day in traffic, and only the airport run is truly awkward.

New York is one of the great places to shop, from the Fifth Avenue flagships and the luxury houses of Madison Avenue to Bergdorf Goodman, Saks and Macy's, with independent boutiques in Nolita and records and young designers in Greenpoint.

Mexico City is best for things you cannot buy elsewhere: Oaxacan pottery, alebrijes and lacquerwork at La Ciudadela, Saturday art in San Ángel and the antiques of La Lagunilla. Polanco has its luxury strip, but fashion is not why people come.

The gap here is wide. In Mexico City a market lunch, a metro ride and a car across town all cost little. Only the boutique hotels and fashionable restaurants of Roma, Condesa and Polanco come close to European prices, and staying in the Centro keeps the whole trip cheap.

New York is among the most expensive cities in the world to visit. Hotel rooms are small and dear, every bill carries tax and a tip, and the big sights charge high entry. Free parks, the High Line and rooms in Brooklyn or Queens help, but it never becomes a cheap trip.

New York is in a class of its own. Broadway and Off-Broadway, the Metropolitan Opera, the Philharmonic and City Ballet at Lincoln Center, Carnegie Hall, and a century of jazz in Village basements and Harlem fill every night, almost all of it in English.

Mexico City has a solid formal programme at Bellas Artes, where the Ballet Folklórico is the easiest show for visitors without Spanish, though most of its theatre is in Spanish. Its most memorable nights are less formal: mariachis in Plaza Garibaldi and the noise of lucha libre at Arena México.

Neither is a restful place, and in both you rest by getting away from the noise and crowds, though in different ways. Mexico City's own ritual is the temazcal steam bath. For a real retreat people head south to Cuernavaca and Tepoztlán, and in the city calm means Coyoacán's shaded plazas or a bench in Condesa.

New York answers with bathing: the old Russian and Turkish bathhouse in the East Village, the banyas of Brighton Beach and the Korean spas in Queens, plus slow mornings in the Brooklyn Botanic Garden or at the Cloisters.