You should not use libcxxabi directly. To my understanding it is a kind of platform abstraction library, providing low level functions needed to implement libcxx.
If you are asking about using libcxx or libstdc++, the differences are mostly the license, newer standard version completeness (the clang project seems slightly faster in implementing recent C++ revisions) and the fact that you have two alternative implementations.
There seems no pressing reason for one of those above the other. I would stick to the one that is better supported on your system. Both projects aim at being ABI compatible, so it should be possible to use either without any breakage.