As far as I know, unless its on one booking all the way through then neither airline will take responsibility and rebooking on Etihad for the next day may cost you more than when you originally booked (depends what Etihad's policy is as to whether they will just transfer you on the same fare basis or if you need to pay to upgrade to whatever fare basis is left on tomorrow's flight).
Of course neither will be liable for any extra costs incurred such as accomodation.