These are two implementations of the MPI standard . In the late 90s and early 2000s, there were many different implementations of MPI, and the developers began to realize that they all invented the wheel; there was a bit of consolidation. The LAM / MPI team joined the LA / MPI, FT-MPI and, ultimately, PACX-MPI teams to develop OpenMPI. LAM MPI stopped developing in 2007. The code base for OpenMPI was completely new, but it brought ideas and methods from all different teams.
MPI : OpenMPI MPICH2.