LibreOffice has technical advantage because it is released under a copyleft
license, unlike OpenOffice. Useful code can be copied from OpenOffice to
LibreOffice but not vice versa. Thus LibreOffice contains all useful features
of OpenOffice and more.
Both licenses are free software but only copylefted LibreOffice is guaranteed
to remain so. OpenOffice (forks) can turn proprietary.
OpenOffice is certainly a better known name among the general public but the
very strictly regulated nature of the development under Sun Microsystems has
caused much bad blood associated with the name in the developer community.
This will likely change. Now e.g. Debian and all it's derivatives, Trisquel
included, come with LibreOffice.
You might want to read the whole history of the office suite and it's various
reincarnations, it's quite interesting actually.
