Functional testing doesn't change much between on-prem and cloud software: a feature either works or it doesn't. What changes is the set of risks that sit underneath the features, and those need their own attention.
In a shared cloud environment, one tenant's data, load, or configuration should never leak into another's. Testing needs to specifically target tenant isolation, not just assume it holds because the feature works for one account.
An on-prem deployment often has predictable load. A SaaS product can see sudden spikes from any customer, at any time. Performance and load testing stop being occasional checks and become a standing part of the test strategy.
Cloud products typically depend on third-party services: payment gateways, authentication providers, external APIs. Testing has to account for those dependencies failing, being slow, or changing without notice, not just for your own code behaving correctly.
Continuous deployment means regression testing has to run constantly, in small increments, rather than once before a quarterly release. This is usually where automated regression testing stops being optional.
Where on-prem software often runs in a fixed environment, SaaS products need to account for a wider range of browsers, screen sizes, and integration points, since you don't control what the customer connects it to.
Talk to our QA team about cloud and SaaS testing.