Broadly neutralizing antibodies (bNAbs) that target the membrane-proximal external region (MPER) of HIV gp41 envelope, such as 4E10, VRC42.01 and PGZL1, can neutralize >80% of viruses. These three MPER-directed monoclonal antibodies share germline antibody genes (IGHV1-69 and IGKV3-20) and form a bNAb epitope class. Furthermore, convergent evolution within these two lineages towards a 111.2GW111.3 motif in the CDRH3 is known to enhance neutralization potency. We have previously isolated an MPER neutralizing antibody, CAP206-CH12, that uses these same germline heavy and light chain genes but lacks breadth (neutralizing only 6% of heterologous viruses). Longitudinal sequencing of the CAP206-CH12 lineage over three years revealed similar convergent evolution towards 111.2GW111.3 among some lineage members. Mutagenesis of CAP206-CH12 from 111.2GL111.3 to 111.2GW111.3 and the introduction of the double GWGW motif into CAP206-CH12 modestly improved neutralization potency (2.5-3-fold) but did not reach the levels of potency of VRC42.01, 4E10 or PGZL1. To explore the lack of potency/breadth, viral mutagenesis was performed to map the CAP206-CH12 epitope. This indicated that CAP206-CH12 is dependent on D674, a highly variable residue at the solvent-exposed elbow of MPER. In contrast, VRC42.01, PGZL1 and 4E10 were dependent on highly conserved residues (W672, F673, T676, and W680) facing the hydrophobic patch of the MPER. Therefore, while CAP206-CH12, VRC42.01, PGZL1 and 4E10 share germline genes and show some evidence of convergent evolution, their dependence on different amino acids, which impacts orientation of binding to the MPER, result in differences in breadth and potency. These data have implications for the design of HIV vaccines directed at the MPER epitope.