The basic difference
The Test Case gives you of the test while the Test Run includes information related to running the Test Case. The Test Case will include the test steps, conditions expected result. A Test Run will keep track of the test when the test , Test Run status, Not Run, In Progress, Passed or Failed, Assignee and the Test Cycle the test run lives in.
Example of a Test Case:
Example of a Test Run tab of a Test Case:
Example of a Test Run:
Test Run tab example:
Note: please be aware that it is a common misconception that all fields in Test Cases will be replicated in Test Run. Only the Name and Description of the Test Case will be inherited by the Test Run.
The connections between them
A Test Plan is a container of all the Test Cases you intend on running. When you create a test plan you will have to add test cases to the Test Groups.
You can create Test Cycles in your Test Plan to organize the Test Cases that you want to run in different stages of your project. For each Test Case that you add to a Test Cycle, a Test Run will be created.