A lot of the other resorts will sell day passes in the summer. I'm not overly familiar with the Biltmore but I'm pretty sure that they don't have water slides like the Pointe South Mountain or the Flowrider like the Westin Kierland. Those are two things that the teens may want to do for a day.
And a big +1 on the idea that if you go for a hike, go early, and I mean early. Please don't be one of the idiots that has to get plucked off of Camelback by Phoenix Fire.