If you're using Windows, use OpenOffice. If you're using Linux, use LibreOffice.
OpenOffice was stagnating a while ago, so I switched to LibreOffice. Both systems were a little sloppy back then, but LibreOffice seemed a little nicer. When OpenOffice was revived as Apache OpenOffice, I tried it out, but it seemed to have poor support for file formats and it couldn't import some of my old LibreOffice files (even though they're both supposedly using the same standard ODF file format), so I stayed with LibreOffice.
But LibreOffice has always been problematic for me on Windows with lots of features not working quite right for many years. I just tried OpenOffice right now, and it just "works." No rendering artifacts. No problems with PDF export. It just works. The download and installation experience isn't quite as polished as with LibreOffice, but it just works.
I think the LibreOffice developers are mostly Linux developers who work for Linux distributions, so it is just works better there and is better integrated with those distributions. OpenOffice is used in commercial Windows products, so the developers make sure it works properly on Windows.
Update (2018-3-10): Development on OpenOffice has mostly stopped, so I decided to try the latest LibreOffice 6. PDF export was still broken on Windows. Performance on Windows has now gotten so bad that it was painfully slow to just type up some text on some slides for a presentation. I went back to OpenOffice.
