Some people belong to both, others prefer one or the other.
On LDO's message board you do not have to give your real name, which can be a benefit if you do not want to share your information about becoming a donor. We also moderate these pages. The messages here are organized in a way that you can find topics easily, and you can search for topics or keywords.
On facebook you would be using your name, and that's fine if it is not a problem for people. Users have to abide by facebook rules and guidelines. Each topic thread gets pushed down as newer posts are made, and searching the group for topics is not an option.