Wouldn't it be very unusual for a public school to have a music program rigorous enough to challenge a Julliard-level musician? Have you confirmed the quality of this program? I would not give the music curriculum any weight unless you have done so. I would love to know where this is, if you would be willing to share.
With regards to the swimming - is he already on a team? In my experience most very competitive HS swimmers compete outside of their schools (I could be wrong on this). If he is not already on a year-round team, is it an option in your area? What are his expectations for the swim team and what does he want to get out of it?
You may already have a lot of information on the swimming and music options that you didn't include but, if not, I would definitely do more research before going further in the decision-making process. It seems like some of the "pros" of the public school option might not really be that significant.
