Anecdotally speaking, I've taken Uber quite often in Austin, as well as various cabs, and they always seemed comparable to me. In fact, if you take into account Uber's surge pricing (which cabs don't have, afaik), Uber came out more expensive on average.
Where I'm from (Toronto) the taxi rates are fixed by law and the Uber rates are publicly available on their website.
Taxi rates: $3.25 + $1.74 per km + $0.50 per minute
UberX rates: $2.50 + $0.80 per km + $0.18 per minute
It's not even close.
My house to the airport via taxi... $45-65. 30-40 minutes.
I fly 2-3x per week, thinking about putting together some expense reports to back this up... but for now you'll have to take my word. (=
That comparison is only helpful if you note how long it took for both to show up.