Presumably because the black cabs have had far more stringent background checks run on them. I think anyone can download the Uber app and become a driver, but even if that isn't true, my perception is that it is probably a lot easier to get round these checks.
You can't, and never could "just download the Uber app and become a driver".
Here is the process for London, which does seem pretty thorough to me. Crucially, it does also involve some TfL licensing accreditation, so I'm not sure why the 'blame' falls back on Uber.
Step 1 - Visit ignition centre
Visit our Ignition centre anytime during the opening hours. You'll get FREE personalised help with every step of your TfL private hire licence application.
Step 2 - Complete Enhanced DBS
We can help you send off for an Enhanced DBS check as part of your application process. Drop in with your passport, driver’s licence and a bank statement (from within 3 months).
Step 3 - Get Medical Check
Call your GP to ask if they can complete the Private Hire Medical Form, check the next availability, cost and inclusion of the eye test.
Step 4 - Submit Private Hire Application
Receive in-person support at our Ignition centre to submit your private hire application to TfL. Drop in with the required documents (Medical, Enhanced DBS, passport, driving licence, NI number and debit/credit card with £250).
Step 5 - Prepare for Topographical Assessment
Once you’ve submitted your PCO application, you will be invited to sit for a topographical test with TfL. Uber offers free preparation sessions so make sure you come along!
Step 6 - Start driving with Uber
Once you’ve received your private hire licence, take a look at some vehicle rental options offered by third parties. Make an appointment at our Greenlight Hub to onboard as a new driver partner and get on the road!
Source >>> https://www.uber.com/en-GB/drive/london/get-a-license/london-ignition/