A use case is written from a "user" perspective describing the interaction of a piece of software between the user and the software.  These are written in common language typically from the business or user point of view and in enough detail for the developer to create a piece of software. Typically written in a MS Word type tool.  Use cases capture the functional requirements of the system.  It describes the expected interaction the user will experience, in detail.  The audience is the business, for signoff, and technology for development.

A Test Case is written using the use cases for a source.  It takes a use case to a deeper level so that software testers can exercise every possible scenario that could occur, negative and positive scenarios.  One Use Case can turn into 10 test cases.  10 test cases make up a test script.  Typically Test Cases will be written in a testing tool like Test Director, but also can be written in MS Word.  The audience is QA testers.