I never understood the difference between Mainframe Computers / Computing and Cloud Computers / Computing.

In both cases, a centralized server connected to the internet does all your application, storage and processing. If that server goes down / connection is lost, in both cases you and your "SaaS" applications are SOL ( Sh*t Out of Luck). Also in both cases, you offer hackers a centralized location to target and hack.

What's the difference, except in the name and in the marketing ?