To be born is to be given birth to, as babies are born. Or it can mean "to be created": ideas are born the moment we think of them. It also means "to arise from": Timmy's stomachache was born of wolfing his food.
Borne is the past tense of bear, in the sense of "carry." To be borne is to be carried: a mosquito-borne disease; or to be endured: Timmy's stomachache had to be borne until it finally went away.