The short answer
Stay on the Upper East Side if you want to sleep at night and still get to a Broadway curtain in about half an hour. Stay in Times Square if the shows are the whole point of the trip and you'd rather walk five minutes home after one than take a train.
That's the real trade. Everything else people argue about (which is safer, which is more "New York") matters much less than whether you want the noise outside your window or twenty blocks away from it.
We're at 341 East 62nd Street, so we're not neutral. But we send people to Times Square hotels all the time when it's the right call, and we'll tell you below exactly when it is.
What the two areas are like after 10pm
The Upper East Side goes quiet after about nine, and Times Square doesn't go quiet at all. The side streets in the East 60s are residential: townhouses, a few small apartment buildings, doormen out front, dog walkers, the occasional taxi. The avenues (Lexington, Third, Second, First) keep some traffic and late food, but you're one turn off them and it drops away.
Times Square runs on a different clock. Theatre crowds empty onto the sidewalks around 10:15, the buses on Eighth Avenue don't stop, and the signage is bright enough to read by at 2am. Hotels there have good glazing and blackout curtains for a reason. If you're a light sleeper, or you're travelling with a kid who goes down at seven, that's the thing to weigh, not the map distance.
Getting to a Broadway show from East 62nd Street
It's three stops on the Q. The Q at Lexington Av/63rd St runs to 57 St-7 Av, 49 St, then Times Sq-42 St, which puts you inside the theatre district with a few blocks to walk to most houses. Figure roughly 25 to 30 minutes door to seat if you leave in good time, plus a few more if you're walking west of Eighth Avenue.
Cabs and rideshare take about the same crosstown at night and longer at 6pm, when everything from 42nd to 57th is stuck. Our honest advice for an 8pm curtain: eat near us at 6, get on the train at 7:10, and you'll be in your seat with time to spare.
Getting to Central Park, the museums and the shops
The Upper East Side wins this one outright. Central Park's Fifth Avenue entrance at 60th Street is about an eight-minute walk from our door, straight west along the quiet blocks. Museum Mile starts at 82nd and runs north up Fifth. Bloomingdale's is at 59th and Lexington, and Madison Avenue's shops start a couple of blocks west of us and run north.
From Times Square, Central Park's south edge is a fifteen-minute walk up Seventh or Broadway through the busiest sidewalks in the city, and the museums are a subway ride. It's doable. It's just never quick.
What you give up on the Upper East Side
You give up the walk home from the theatre and some of the late-night food. After midnight up here your options thin out to a handful of delis and diners on the avenues. Times Square and Hell's Kitchen keep going. If your trip is built around 10:30pm dinners after shows, that's a genuine argument for staying west.
You also give up the feeling that you're standing inside the postcard. Some people want that on a first trip, and there's nothing wrong with it. Go stand in it for an evening, then come back east and sleep.
How we're set up, for what it's worth
Ours are apartments, not hotel rooms: studios up to the whole house, each with a kitchen, and no lobby or key card to deal with. Your door code arrives before you get here and you come and go as you please. That matters more on a longer stay, when eating out three times a day gets old and you want a fridge and a washing machine's worth of normal life.
A family of four in one apartment with two bathrooms is a different trip than a family of four in two connecting hotel rooms off Eighth Avenue. Both work. One has a kitchen table.
Who should pick which
Pick Times Square if: you have shows three nights out of four, you're here for two or three nights only, you want everything within walking distance, and noise doesn't bother you.
Pick the Upper East Side if: you're staying four nights or more, you're travelling with kids or a dog, you want Central Park and the museums on foot, you want to cook some of your meals, or you simply want the street outside to be quiet when you turn the light off.