We have not stayed at either resort, but we were debating the same thing for this August. We're booking the Free Dining promotion, so the price difference was negligible, it was just a matter of availability and playing with the dates. We're still flexible by a few days either way on our trip, so we were able to find dates that worked for both. We always thought we'd prefer Beach Club, and at one point we were leaning towards doing the dates that worked with Beach Club.
However, we have decided to just do Yacht Club either way. The big reason, and it's been mentioned, is the balconies. We noticed it in pictures, then looked it up, and found that many Beach Club rooms have "standing balconies," whereas Yacht Club has full balconies on all their rooms.
The balcony is very important to us because we also have younger kids (our son will turn 7 during the trip and our daughter will be 5). When the kids are sleeping, the balcony is our place to go out and relax, have a drink, and talk, without worrying about disturbing them. We still intend to try to get them to nap this trip, so the same thing then. If we don't have a balcony, we're stuck sitting in the room with them. It's a big reason we started doing Deluxe two trips ago, and we call it our sanity place - just our brief break from the kids. Since you have young kids as well, this is something to certainly consider.