Test Director is supposed to be a an encompassing application that helps with all aspects of testing from test case management to down to running your automated tests. It then reports on all of this.
If you have a lot of money then it may be the right option for you.
Selenium is a UI test automation framework that is people use instead of HP Quick Test Pro (QTP). QTP, like Test Director, costs a small fortune to setup and get it going. Selenium is very good at testing at AJAX applications where QTP can cope with it. Selenium runs in the browser so doesnt rely on UI maps and doesn't need to have a object store for the page. The Page Object Pattern for UI testing can handle this in a more intuitive manner. With Selenium 2 (now in alpha) there is going to be better support for more native actions like key strokes which QTP is slightly better but once it has fully released then I feel Selenium 2 will be the best Web Test Automation framework out there. QTP will still be able to do desktop applications but with more and more applications going into Browsers it will have less and less marketshare of all applications out there.