Do you like pool time? If so, Beach Club. In our experience over multiple stays, Beach Club's feature pool, Stormalong Bay, is less crowded than the Polynesian's feature pool. At Stormalong Bay, CMs stand at the entrances & scan bands to make sure only resort guests use the pool. At Polynesian, there is only a gate that crashers & the friends/family of onsite guests can easily get through. And the resulting difference in pool crowding has been noticeable to us. Not to mention that Stormalong Bay is the best pool complex at Disney World, hands down.
Beyond pools, it's a toss-up. They're both lovely resorts. The Polynesian has better casual & family dining options, in my opinion. If you're staying at an Epcot area resort during the reservation period, when you can't just drop into Epcot at will to get food, your dining options might feel a bit more limited than at Poly, which has very good dining, as well as monorail access to two other resorts that have good dining.
Beach Club rooms are all under one roof, so no need to walk outside to go between your room & the lobby or a resort restaurant, unlike at Poly. That really helps when it's raining, which it does often at WDW. Second floor Polynesian resort rooms (not referring to villas) have no balconies, whereas all Beach Club resort rooms have either a patio or a balcony, although some of their balconies are small.
Polynesian is on the monorail line, which gives great access to MK & Epcot, but Beach Club is walkable to Epcot & Hollywood Studios, so it's really a matter of which parks you spend more time at, in terms of which resort is more convenient. The Polynesian offers a lot of views of MK & the castle from different spots around the resort. For MK fans, the Poly is hard to beat.