Test design: ---
Documentation specifying the details of the test approach for a software feature or combination of software features and identifying the associated tests.
A Test Automation Framework: ---
is a set of assumptions, concepts and tools that provide support for automated software testing. The main advantage of such a framework is the low cost for maintenance. If there is change to any test case then only the test case file needs to be updated and the Driver Script and Startup script will remain the same. There's no need to update the scripts in case of changes to the application.
Difference between them is that test design is documentation of testing approach where as automation framework is actually implementing it using testing tools.