I saw Phantom this past summer for the first time, and it had more charm than any musical I've seen yet. I haven't seen Heights or August but have herd nothing but great and wonderful things about both shows.
I would say August. Depending on how often you visit NYC, I think you can see Phantom and Heights some other time.
Broadway Shows I've Seen: Hairspray, Chicago, Little Shop of Horrors (2003), The Wedding Singer, Spamalot, Riverdance, Rent, Beauty and the Beast, Spring Awakening, Wicked, Legally Blonde, Phantom of the Opera, Sweet Charity (revival), Drowsy Chaperone, The Lion King, Dreamgirls(2010 Tour).