Within an hour of landing at La Guardia, I was eating a pizza in Little Italy. Brussels sprouts, caramelized onions, red sauce, mozzarella. The dough was delicious, but the pizza didn't come together. It was the first of many. I'm not sure I really made my peace with New York style, but I can certainly appreciate many aspects of it.
To start, I should mention that most pizza eaten in Chicago isn't thick or stuffed or Chicago style. If you are at a party or just want to watch a movie at home, you get a thin crust. Because you are paying a bit for delivery, you might as well get a medium, even if its just you - so you'll have leftovers.
The crust will be a little crispy, but still flexible. The cheese a solid layer, a dry mozzarella base, cooked at least partly golden. The crust peeks out from the edges, but is only a little thicker there. It might be cut into squares - if you ordered a large it almost has to be.
There are definitely restaurants to go and sit down for pizza, where it is more likely to be Chicago style, or Italian, boasting brick ovens, etc. Still, pizza for Chicagoans, most of the time, is a usually a delivery comfort food.
In New York it’s different. Pizza's mythos is as a street food, bought by the slice, eaten with one hand on the way to wherever you are going. The design of a New York slice lies in this purpose.
Squares won't work, it has to be one big piece to hold. You don't want to get your hand messy, so the edge crust should be thicker, like a handle. If you want to hold it with one hand, it can't be flopping around, so there are two choices: cardboard stiff, or so flexible you can easily do what to a Chicagoan seems a blasphemy: fold the pizza lengthwise, holding the thick crust side in a "V" shape, giving it a rigidity to hold horizontally, keeping the ingredients stable. Also as you take bites, your lips only touch the crust, so you end up with less sauce and fat on your face. Handy if you are walking down the street. To make this happen, the crust is thin and floppy, undercooked to a Chicagoan.
Aside from function, I'd have to say that the pizzas I had in New York were more ingredients oriented. The sauce was always bright and distinct, the dough sweet, cheese often fresh mozzarella - leading to less browning.
I think typical pizza in Chicago doesn't have the same attention to individual ingredient quality, but develops a lot of flavor in the execution - the browning of the crust, it’s slight crisp texture, the golden wonderful in the cheese.
I'm not interested in dogmatic arguments about what the real pizza is. Let it be constantly reinvented! Having had pizza in DC and San Francisco you are forced to be a little more open minded.