A couple things, in order to drive for Uber the vehicle must be model year 2002 or newer. In some cities they have to be even newer than that. Given the state of some taxis I've seen, I wouldn't hold them up as a beacon of vehicular safety.
Also, they (Uber, anyway) have $1 million liability policies if the accident is the fault of your driver and you're injured while using their service. Yes, all cabs carry coverage but it's generally the minimum required and good luck dealing with those companies.
Uber and Lyft have their issues, but if I have to choose between them or a cab company, I'm going with the ride sharing service every time. My worst experience with them is still an order of magnitude or two better than my best experience with a cab company.