The advantage of using a framework is two-fold:
- If using a theme framework to build a regular theme, it can give you a head start by giving you mature code (in most major frameworks, anyway) that you can use as a base for your customizations. This gives you more control of the theme framework code but also creates a bit of a problem: you now have a forked version of the theme and would need to run a comparison tool to incorporate updates from the original theme author.
- If using a theme framework as a parent theme, you can build a child theme that sits on top of the theme framework, allowing you to add your own code or remove things programmatically using actions and filters from the parent theme, without touching the parent theme's code. You allow them to continue developing the theme framework and update to the latest version to avoid having to maintain your own forked version of theme code that you did not write.
There are quite a few different theme frameworks out there. The only ones I've ever used consistently are the UpThemes Framework (which is a theme options framework, not a full theme) and Thematic which has not been updated in quite some time. Obviously, the Woothemes Framework, Genesis, and Hybrid Core are pretty popular ones. I think Genesis and Hybrid Core are great options because of the support packages that come with them and the documentation you get.