> depending on where you live there is no guarantee a taxi is around the corner
Exactly. You go to a warehouse party in East Williamsburg and it's 5 AM and feel like going home. The public transit situation in Brooklyn would make that trip take a while (wait 20 minutes for the L, get off, wait 20 minutes for the G, or alternatively wait an hour for the next bus). And what kind of yellow cab is going to hang around an industrial district waiting for a fare? Uber wins every time.
It's the density thing. The taxi model works well enough in Manhattan because there are enough people around the hot spots to provide off-the-street fares. A lot of things that work in Manhattan don't always transfer easily to the lower-density reality of the rest of the city/country.
For one it usually costs less. And two, depending on where you live there is no guarantee a taxi is around the corner. Uber is more convenient.