They're pronounced the same. There's also the surname "Bourne".

No. Borne and born are always homophones, regardless of the accent. The vowel sound would vary according to the speaker's regional pronunciation, and the 'r' sound would be pronounced in some accents (for example, most North American, Irish and Scottish accents) and not in others. However, as the final 'e' is silent, no native speaker would make a distinction between the two words.

Depends on which accent you are looking for.