Whether or not they fill the same requirements, they made it so easy to get a car that it was easier than standing on a street corner and hailing a yellow cab.
And aside from that, they added accountability. Yellow cab drivers were rude, the cars were moderately clean at best, and generally took opportunities to rip people off. I remember leaving my phone in a cab around 2008, I immediately called it and the dude picked up, said he was around the corner and to meet him outside. He was more than willing to give me my phone back- for $60 and had a shit eating grin on his face. I had the same thing happen in an Uber about 5 years later- the guy dropped the phone off at my house free of charge with a smile.
There is a reason that NYC taxi medallions have pretty much halved in cost since Uber started here- because they are directly competing with yellow cabs!