The rift goes back as far as the 1940s. That's when Chicago or 'deep dish' style pizza was invented. New York-style pizza had been around since the early 1900s.
The two styles couldn't be further apart.
New York-style pizza is made from being hand-tossed, which results in thin crispy crust. Some even say the NYC water supply, which is high in mineral content, also adds to the unique New York style. The slices can be folded and eaten on the run. This style of pizza is a part of American folklore. In fact, Lombardi's pizzeria, the first in America, is still around today and cooking up pies.
Then you have Chicago-style pizza. There is debate as to the originator, but everyone knows a Chicago style deep-dish pizza when they see it. The pizza is cooked in a pan, similar to a cake pan, which gives the pizza depth. Whereas you would lose all self-respect if you eat a NY-style pizza with a knife and fork, nobody will judge if you dig into a Chicago-style pizza with utensils. In fact, utensils may be necessary, since there is often a thick layer of toppings. The longer baking time is why the "toppings" are often put below the cheese, protecting them from getting burnt.
Critics say Chicago-style pizza isn't even pizza... it's a casserole. But maybe they are just haters because they know deep down Chicago-style pizza's excess of gooey cheese makes it taste better.
New York started it all, but some say Chicago perfected it.