The short answer
Greenwich Village wins on nights out. The Upper East Side wins on mornings, space, and sleep. Pick based on which half of the day your trip actually happens in.
That sounds glib, so here's what it means on the ground. The Village is dense in a way that's hard to explain until you're standing in it: crooked streets that don't follow the grid, a restaurant or bar every twenty feet, Washington Square Park full of people at midnight, and six subway lines under West 4th Street. You never plan an evening there, you just walk out and find one. The cost is noise, small rooms, and paying a premium for both.
The Upper East Side is the opposite trade. Above 60th Street the blocks go quiet after nine, the sidewalks are wide, and the buildings are low enough that you can see the sky. Central Park is at the end of the block for a good part of the neighborhood, Museum Mile starts at 82nd Street, and you get noticeably more square footage for the same money. What you give up is spontaneity at night. Dinner here is a plan, not an accident.
Where we sit in this
We're at 341 East 62nd Street in Lenox Hill, which is the southern edge of the Upper East Side and, honestly, the part that argues best against the neighborhood's sleepy reputation. The F and Q at Lexington Avenue and 63rd Street are around the corner, and the Q gets you to Midtown in minutes and down the Second Avenue line without a transfer. The Fifth Avenue and 60th Street entrance to Central Park is about an eight-minute walk. The Roosevelt Island tram is two blocks away. Bloomingdale's and the shops on Lexington and Third are right there, and Second and Third Avenues in the 60s have plenty of places open late, just not the wall-to-wall density of Bleecker Street.
Our apartments run from studios up to the whole house, all with full kitchens. That's the other half of the comparison people forget: in the Village you're usually booking a small room, and here you're booking a place with a table to eat at and a door of your own. No lobby, no key card, just your own front door.
Side by side
The table below is what we'd tell a friend deciding between the two.
When to choose Greenwich Village
Choose the Village if it's your first trip and you want to be in the middle of things, if you're 25 and out until 2am, if jazz clubs and comedy cellars are the point of the trip, or if you're spending most of your time downtown and in Brooklyn. It's also the better base if you hate planning. The neighborhood does the planning for you.
When to choose the Upper East Side
Choose the Upper East Side if you're traveling with kids, if you're here for more than four or five nights, if you want the park in the morning and the Met in the afternoon, if you're working while you're here and need quiet during the day, or if you've done the loud version of New York already and want the version people actually live in. Long stays especially: a month in the Village is expensive and tiring, a month up here is a life.
Midtown's best blocks are the quiet ones. Same goes for the East 60s.