Hi - I'm one of the people that uses Uber and loves it. I live in a big city, Chicago, and guessing from your statement you don't live in a taxi city. Let me tell you why I love uber and hate taxis (or why I only take Uber Taxi).
1) I used to travel for work, and for me that meant getting a 5:45am cab ride from downtown Chicago to the airport. Now, I'm sure you're aware that Chicago is freezing during the winter months, so standing out in the snow at 5:45am on a monday is not something I want to do. What were my options before Uber? 1) Wait outside and try to hail a cab. 2) Call dispatch and be told that a cab may or may not be on its way. This is the real pain that Uber solved for me. I cant tell you how many times I called and was told a cab would be there in 5 minutes, then 15 minutes later, no cab had showed up. So now, its 5:55am on a Monday in the cold Chicago winter, I'm stressing about missing my flight, all because a cab couldn't show up when I ordered it. Uber solved this exacty pain point that I felt. On another note of why I hate taxis and only take uber taxi. A few years ago when Chicago was going after the olympics, they required all taxis to accept credit cards. Now, try to pay for a fare in Chicago with a cab, 50% of the time 'The machine is broken', but when you have no cash, it magically starts working again. I get that these guys do not want to pay the 5% charge to process it (another issue is cab companies upcharging to process cc, but not for this discussion). With uber, I never have to worry about the driver not able to make change for me and just expects hes getting a bigger tip, I never have to worry about only having a CC on me and the driver bitching and complaining about accepting it. Uber solved this problem. Now, they haven't just solved my problem, they've also solved their drivers problem. Talk to a cab driver, ask them how they get paid for their credit card receipts. I'll be willing to bet that they have to go to a special office, receipts in hand, wait a few hours in line, all to get paid. That is ridiculous! Uber direct deposits the money in the drivers bank account, no time wasted by waiting in line, no worry about lost receipts. Now for my last point, imagine you left your phone/wallet/camera/foobar in a taxi in Chicago and you paid in cash. Try calling our 311 number and reporting it, you have about as good of a chance of finding it as comcast does at providing great customer support. With uber, you know your driver, you can simply contact uber and or the taxi company to give specifics on what you lost and where you lost it.
tldr: I <3 uber and Uber > taxis