What is the difference in meaning (however slight) between the two following sentences?
Both brothers fell in love with a girl living next door.
Both brothers fell in love with the girl living next door.
What is the difference in meaning (however slight) between the two following sentences?
Both brothers fell in love with a girl living next door.
Both brothers fell in love with the girl living next door.
It's an extraordinarily-slight difference.
I would suggest that 'a girl living next door' implies that she really is literally living next door, while 'the girl next door' or 'the girl living next door' implies that it might be meant in a more-figurative sense.
It's also theoretically possible that both brothers could fall in love with a girl next door but not actually the same girl, but that falls solidly into becoming a riddle, in my opinion.