The part I really disliked about the NYC cabs is that you never knew how much it going to cost you. And similarly the driver never knew where they would have to go. With Uber, both parties have this information ahead of time.
I went to NYC for the first time about 20 years ago. I was looking for a cab from JFK to midtown. A limo driver said he'd take me for $30 (more or less -- I don't exactly remember). I had no idea really if that was a good price, but I remember thinking, "at least I know what the price will be," and I took his offer.