FAQ/Difference between LibUIDropDownMenu and NoTaint_UIDropDownMenu
They are mostly the same or says identical (at least in version of 7.0.2.24015).
The main reason to create LibUIDropDownMenu as a separate (or says new) library is due to NoTaint_UIDropDownMenu has been created for years, and has been used in many addons. But that library was also outdated for years until recently.
The main problem is many addons are still using the outdated library and some of them were also abandoned. While still there are a few addon authors are keeping their addons up-to-date and updating with the latest of NoTaint_UIDropDownMenu, the outdated ones will keep throwing errors and it would be hard to ask all other addon authors to update NoTaint_UIDropDownMenu.
Therefore, a newly created library to replace NoTaint_UIDropDownMenu should make things easier.