In almost all back stories, Mothra is shown as an egg which hatches into a larvae. In King Of The Monsters (hereafter abbreviated KOTM), this egg was discovered in a temple and has been monitored by the organization MONARCH for several years. In the original Toho appearance, the egg had simply washed up on the shore in Japan.
In the original Toho appearance, Mothra was a dietydeity. A mystical, divine being. In KOTM, Mothra is a prehistoric insect-like species that lived during the same time period as Godzilla.
As far as powers, they're somewhat consistent. The larvae stage uses silk to incapacitate enemies, while the imago version possesses an abdominal stinger, forelimbs for close combat, wings that generate wind, and also silk. Being based on a moth, it's also fragile and susceptible to heat, which are some pretty glaring weaknesses. One major difference is that in her Toho films, she could fire poison darts. In KOTM, that was replaced by a single abdominal stinger.