The difference being: London saw a need for cab drivers and introduced licensing to ensure quality, New York (and other cities) saw a need for cab drivers and saw a revenue stream.
It's also very nice to be able to get in a black cab and say something like "The Civil Service Club please" and they'll know exactly where to go, no prodding at a phone/SatNav for a minute or asking for a postcode.
The downside is that some cabbies use that same knowledge to their benefit. I've been in several cabs where the driver has purposely used a route that's bad for traffic (rather than the obvious alternatives) so they can earn more money for a journey. (This is more common at times when fares are sparse.)
And then there was one time when I dozed off as we were crossing the Williamsburg Bridge. I woke up about 20 minutes later and we were somewhere near Coney Island (I lived in Bushwick, exactly a mile from the bridge. Coney Island is the other side of Brooklyn). I told him he could take me where I originally said for half the price on the meter, which was still more than the usual fare, or let me out without incident. That was a painful subway ride home that morning.
Plus, they can always become Uber drivers, and have better earnings and make their own hours. 90% of all drivers I've had are happy to be driving for Uber.
The taxi system is — or was — indicative of all these weird east coast attitudes that I, as a westerner, find completely insane. All these awful inefficiencies and shitty attitudes baked that everyone accepts as normal because that's just how things are done. (Basic traffic enforcement is another biggie… the "Masshole maneuver," and New Yorkers blocking intersections or parking on sidewalks, landlords who would rather risk lawsuits than do inexpensive repairs…)
Although I have moved on from Uber for various reasons, I will be forever grateful to them for tearing down this horrible system. It sucks that some independent medallion owners are going to take a bath on their loans, but it was a terrible investment in the first place.
I feel for any of the honest cab drivers that didn't pull this kinda crap, but my experience tells me that there weren't many of them.