The verdict
If you need a credible site this week and the shop is a small part of it, pick Squarespace. I picked it for my dad's site because it was so straightforward, and a polished site was live in a few hours. If orders, catalogue size and integrations are the business, pick BigCommerce.
The trade-off on Squarespace is flexibility. You choose from pre-designed sections and cannot reshape the layout freely, and heavy third-party integrations are where you may outgrow it. BigCommerce asks for more effort and gives you APIs and B2B tools in return. A brochure with a few products is Squarespace. A shop that has to talk to other systems is BigCommerce.