WWW::Mechanize is a subclass of LWP::UserAgent, so it comes with all the functionality of UserAgent, plus its own additional functionality. What it is good for is acting like a web browser, while providing easy methods of automating many things that a web browser could do. This includes following links, handling forms, submitting forms, and so on.

The WWW::Mechanize documentation has some examples, but to me it made more sense once I dove into starting to use it. Think of a simple task you would like to automate that involves WWW screen scraping. Then sit down with the WWW::Mechanize docs, and figure out how you're going to accomplish this task using WWW::Mechanize (keeping in mind that it's a proper subclass of LWP::UserAgent). Fiddle and tweak, and eventually you'll figure out why it's useful.