A consultant recommends. An FDE ships.
A consultancy produces an assessment, a roadmap and a set of recommendations, then hands the build to somebody else, often a systems integrator or the client's own team. The engagement is complete when the document is accepted.
A forward deployed engineer writes the production code themselves. The engagement is complete when the system is running and handling real volume. That single difference changes what the person has to be good at: less framework, more debugging somebody else's undocumented database at four in the afternoon.