In the wealthy side of Manhattan the city is swarming with cabs. In Brooklyn, Queens, the Bronx, and even Harlem, Washington Heights, etc, cabs can be hard to come by.
UberX is also cheaper than a cab nowadays with the recent price drop.
I disagree with billmalarky below about the experience being better though. Uber's drivers are almost always considerably less experienced than cab drivers. I've lost count of the number of times I've sat in an immobile Uber while the driver futzes with some shitty GPS unit so he knows where the hell to actually go. Also lost count of how many times an Uber driver couldn't find his way to the Manhattan Bridge on-ramp. These are things that are second-nature to cab drivers.
When I get into a cab I might get a surly guy who's yakking on the phone the whole way, but I'll get to my destination quickly, and cheaply. When I get into an Uber I'm almost guaranteed to get someone who is polite, genial, and doesn't really know where they're going. I'll take the former, thanks.
You know, you're right. When someone says "NYC" I pretty much only think of Manhattan (from the top of central park down). I forget how enormous the city actually is.
In Brooklyn and Queens though Uber has been considerably worse than cabs (green or yellow) in my experience. They just don't know the lay of the land as well as cabbies and frequently have to resort to GPS units (themselves not as good as a knowledgeable driver).
With uber you consistently get a pleasant experience.
Quite literally the EXACT driver in the EXACT car will come if you use Uber in my neighborhood because my local car service works with Uber. It costs more on Uber than if you call them directly, of course.