Nothing much in my experience.
Actually, the afternoon food service at BC had a bit more assortment to choose from. It certainly isn't substantial--just a variety: chips, salsa, popcorn, pretzels, cookies, maybe a PB&J and soft drinks. Polynesian doesn't offer much more than cookies, candy, and soft drinks. Fruit is always available though!
Our preference remains Polynesian over BC--hands down. While Stormalong Bay is an attraction, the concierge floor is not nearly as convenient to it as the Hawaii building (where concierge returns in a bit over one month) is to its pool. Further, the other offerings at Polynesian e.g., Captain Cooks for late snacks, the staff of Polynesian, and the monorail access to MK and EPCOT make it more desirable to us than BC.
We enjoyed our stay at BC--especially the proximity to EPCOT-- but found the room size lacking when compared to Polynesian and we didn't enjoy the lounge service nearly as well at BC vs Polynesian. We like being able to self-serve our soft drinks, water, etc. without asking for it from a staff member (provided one is nearby). Further, the evening service at BC seemed to be the same night-after-night. At Polynesian the menu rotates on a weekly basis.
When we first began visiting WDW, I was convinced that my preferences would have been the GF, BC, YC type of style. Our kids really wanted to try Polynesian when they were little (after a stay at GF) and we've all been hooked ever since!
Both are great hotels--I'm sure you'd be happy with either.
