The short answer
Pick Tribeca if you're planning your trip around dinner reservations and want a residential block to come home to. Pick the Financial District, where we are, if you want a subway hub under your feet, the big downtown sights on foot, and more square footage than either neighborhood's nightly rate suggests.
That's the whole decision, and it's a smaller one than the internet makes it sound. These two neighborhoods touch. We're at 17 John Street, and Chambers Street, where Tribeca starts, is about a 20-minute walk north or one stop on the A or C from Fulton Street. Whichever you choose, the other one is available to you on the same afternoon.
How the two actually feel
Tribeca is low. Warehouse buildings turned into lofts, wide sidewalks, streets with names rather than numbers, and not much street noise on a weekday. It reads residential because it is.
We read differently. The Financial District is tall and narrow, built on a street plan older than the grid, which is why John Street bends and Stone Street is cobblestone. On a weekday morning the sidewalks are full of people going to work. By seven in the evening they empty, and on a Saturday the whole district is close to silent. Guests who expected Midtown are usually surprised by how quiet it gets. The Financial District is pretty quiet on weekends, and for a lot of people that's the reason to stay here rather than the drawback.
Getting around: this is where we win
The Financial District has better transit than Tribeca, and it isn't a close call. Fulton Street sits a few minutes from our door and carries the A, C, J, Z, 2, 3, 4 and 5, so you're one ride from most of Manhattan without a transfer. The 2 or 3 puts you at Times Square in roughly ten minutes. The PATH to Hoboken and Newark runs from the World Trade Center, a walk away. Pier 11 on Wall Street handles the NYC Ferry, and the Staten Island Ferry, which is free, leaves from Whitehall at the bottom of the island.
Tribeca has Chambers Street and Franklin Street, which is enough for a resident and thinner for a visitor doing three things a day in different boroughs.
Eating: this is where Tribeca wins
Tribeca has the better dinner. Locanda Verde at 377 Greenwich Street, Andrew Carmellini's Italian osteria, is the kind of table people book before the flight, and Bubby's at 120 Hudson Street is the breakfast institution, open from 8am and worth hitting on a weekday to skip the weekend line. Staying up there means walking home from dinner in four minutes.
We eat well down here, but differently. Stone Street is a cobblestone block of restaurant patios two blocks from us, and it's a bar-and-a-crowd sort of evening rather than a quiet dinner. There are food halls in the Oculus and at Brookfield Place for a fast lunch. If your ideal night is a booked table and a short stumble home, that's a genuine argument for Tribeca.
Sights on foot
We're closer to the things people fly here to see. The 9/11 Memorial and One World Observatory, the Oculus, Trinity Church, the Charging Bull, Battery Park and the ferries to the Statue of Liberty and Governors Island are all a walk from our door, not a subway ride. From Tribeca, most of that list becomes a 15 to 25 minute walk south or a train.
Space, and what you're actually booking
This is the part that decides it for families. Our suites are apartments: the smallest is a 340 sqft Classic King Studio, the Classic 2 Bedroom is 750 sqft and sleeps four, the Deluxe 2 Bedroom is 800 sqft and sleeps six, and the Luxury 3 Bedroom is 1,200 sqft and sleeps eight. Every one has a fully equipped kitchen, and the suites marked W/D have a washer and dryer in the unit. A few, like the Grand 2 Bedroom at 1,000 sqft and the Deluxe 2 Bedroom with Private Terrace, have their own outdoor space.
Downstairs there's the Fitness Center with Technogym equipment, the Guest Lounge with an espresso bar, the Laundry Room with free washers, dryers and detergent, and a Conference Room you can book by the hour if the trip has a work half. That's a different proposition from a hotel room in any neighborhood, Tribeca included.
When we'd tell you to stay in Tribeca instead
Honestly: if your trip is two nights, both dinners are booked up there, and you don't care about being near the memorial or the ferries. Or if you want to feel like you live in New York rather than visit it, and you'd rather that feeling came from the street than from the apartment. We'd rather tell you that than have you book us and wish you were on Greenwich Street.
For everyone else, and especially anyone travelling with kids, a group, or a laptop, the argument for down here is the one we'd make: one train hub, everything walkable, room to spread out, and quiet by Saturday morning.