When trying to understand why words are used to describe certain things, it often helps to look at their etymology. For moment:
mid-14c., "very brief portion of time, instant," in moment of time, from O.Fr. moment, from L. momentum "movement, moving power," also "instant, importance," contraction of *movimentum, from movere "to move" (see move). Some (but not OED) explain the sense evolution of the L. word by notion of a particle so small it would just "move" the pointer of a scale, which led to the transferred sense of "minute time division." Sense of "importance, 'weight' " is attested in English from 1520s. Phrase never a dull moment first recorded 1889 in Jerome K. Jerome's "Three Men in a Boat." Phrase moment of truth first recorded 1932 in Hemingway's "Death in the Afternoon," from Sp. el momento de la verdad, the final sword-thrust in a bull-fight.
Moment would appear to come from Latin for "moving power". The works of Archimedes including On the Equilibrium of Planes which contains statics and levers, was translated into Latin by Gerard of Cremona (c. 1114–1187 AD). Therefore it seems likely that Archimedes used "moving power" to describe the effect of a lever in moving a mass on the other end, and being proportional to the product of the applied force and its distance from the fulcrum on the other end.
Turning our attention to torque ;)
"rotating force," 1884, from L. torquere "to twist" (see thwart). The verb is attested from 1954. The word also is used (since 1834) by antiquarians and others as a term for the twisted metal necklace worn anciently by Gauls, Britons, Germans, etc., from L. torques in this sense. Earlier it had been called in English torques (1690s).
And it seems likely that it comes from latin "to twist".
I would conclude that torque was meant to describe a force that twists an object about its main axis, while a moment rotates the main axis. Think of a long cylindrical steel rod where the main axis runs along its center. For objects with no obvious main axis, one person's torque is another's moment depending upon what the main axis is chosen to be.