The difference is in the mentality. When you're testing against a requirement, you're performing a very black-and-white process: either the requirement was met, or it wasn't.
When you're using a Use Case to guide your testing, you're constructing a scenario that describes how a potential end-user might use the product you're testing. This mentality shift helps you take a broader view of the product, hopefully helping you to find a more diverse set of defects.
As an example, a requirement is often defined as:
The application must have a login screen
Whereas a use case is normally defined more like:
A user should be able to log in and view their previous actions on our website
In the second case, since you're following the mindset of a user who has a specific goal, you're much more concerned with how the system will be used rather than "does it tick all the boxes we've been asked to tick"