The process in which differentiating motor neurons in the neural tube acquire the specialized structural and/or functional features of lateral motor column neurons. Lateral motor column neurons are generated only on limb levels and send axons into the limb mesenchyme. Differentiation includes the processes involved in commitment of a cell to a specific fate. [ GOC:cls http://www.ncbi.nlm.nih.gov/pubmed/11262869 GOC:dgh GOC:dph GOC:jid http://purl.obolibrary.org/obo/go/references/0000021 ]