i think AP held on to the tape because at the time of the interview the story was still unfolding and Cosby didn't really say anything in that part of the interview. i think that the whole time they held on to it they were mulling over when and if to release it.

then after
-the NPR interview aired
-a couple more women's stories were highlighted
-JD story was revealed
-the public began to demand a response from Cosby

that portion of the AP interview became relevant and newsworthy. that's why they released it. i don't think they were necessarily trying to shape public opinion (not that the media doesn't do that), i think that they were responding to public demand.