Having existed pre and post Uber, in my country (Australia), here are my thoughts:
Ordering an Uber is far, far quicker and easier than getting a taxi.
Getting an Uber is still far cheaper. In the old days, if you were traveling after 10pm, you’d very often be charged up to $10 from before you’d even closed the door.
That lead to taxis feeling like luxury vehicles. Catching an Uber seems so mundane by comparison.