Apache OpenOffice and LibreOffice are the modern descendants of OpenOffice.org. For the last few years, almost all Linux distributions have included LibreOffice as their default office suite. However, in the past eighteen months, OpenOffice has reappeared, newly organized into an Apache project, and free software users now have the choice of two full-featured suites instead of one.
Users also have the difficulty of deciding between two almost-identical choices. The two diverged three years ago, and while that can be a long period in software development, in this case, the differences are only starting to become obvious. While considerable cleanup has gone on behind the scenes, the feature sets and underlying logic in both has mutated in only minor ways from the days of OpenOffice.org.
Here and there, you can find new features in the individual applications, especially in the Writer word processor. However, most of the differences are at a higher level, in support for formats and fonts, the policy towards extensions, and, most of all, in the efforts to modernize and standardize the interface.