afaict, from a NYC perspective, Uber is much better if you live in Brooklyn or Queens because, unlike yellow cabs, it's actually possible to get one by using the app. I've heard that in other cities, e.g. San Francisco, the cab service is about as bad as Brooklyn. I'm sure that in these areas, Uber is a big improvement over what was there before.
If you live in Manhattan, it's largely a matter of taste. the cars tend to be cleaner and the drivers friendlier, etc., but will cost you ~20% more. There is better tech integration, e.g. the app can be used to split a ride with your friends, and it integrates with paypal/whatever whereas yellow cabs just barely have credit card readers.
Additionally, you hail Uber from an app and meet at a designated spot, whereas yellow cabs by law have to be flagged down from the sidewalk. Personally, i'm not really into Uber, but I live in Manhattan and am probably just sticking with what i'm used to. I'd probably use it occasionally if i lived in Brooklyn. A lot of my younger coworkers use Uber exclusively (and frequently! i wonder how much of their pay they spend just on transportation).
I suspect that what happened is that other cities took NYC Medallion model (which isn't even really appropriate for Manhattan, at least these days) and applied it reflexively for whatever reason. Coming from Florida, I think it's fair to say that Uber is the first real taxi service many suburban areas are getting. The regulations preceded the industry, and so the industry never even developed. Uber basically said "fuck it," made up some silly story about "ride-sharing" (lol), and that was that.