"BigBrand gets it from the same supplier"
I don't think commerce (especially at Amazon) works the way you think it does.
There is a very good chance that when you buy something at Amazon that says it's BigBrand ... it's just a knockoff. Or a fake that has infiltrated another part of Amazon's supply chain. There is big money in lying to customers through Amazon.
Depends what it is. I would trust Nike products from Dick's Sporting Goods physical store and from the Nike store online but not Amazon or any other online retailer. I'd be suspicious of perfume from (say) Giorgi Armani because rip-off artists have heard of it but who would rip off an boutique bland like Phlur?