Easy to get started. The admin dashboard is extremely easy to use.
It's possible to customize just about anything. After using other shopping cart services this is by far the easiest to design a custom site down to the last detail. Very easy to overwrite template and CSS files.
Trying to create a eCommerce site from other platforms while being PCI compliant can be a huge headache. Knowing that the right security is in place is one less thing to deal with and worry about.
When I started using BigCommerce you could use FTP to manage files, now you can only use WebDAV. Which is only a minor inconvenience.
The Options and Options Sets interworkings can be cumbersome with lots of products that need a variety of different options. Having to create an individual option set for a single product that offers different combinations of sizes (like newborn-kids clothes) is not fun.
It's made store operation run more smoothly. Our store hasn't been open for a year yet so it's difficult to provide any numbers, but it's definitely saved us time compared to other shopping cart services we've used.
The inventory tracking is great because we can see what stock is on hand without having to use another service.
With the combination of the ShipStation add-on it's made shipping orders a breeze. No more standing in line at the post office.
Coming from being a Volusion user, I prefer BigCommerce. If you're a site designer it's a no brainer. Volusion uses tables for everything and many elements without IDs and Classes. All elements in BigCommerce are named and easy to manipulate using CSS. And all parts of the template are directly editable using WebDAV or their browser editor.
Well suited when only needing a storefront. If it's needed as just a part of a larger site it would have to be used on a separate subdomain/address. Doesn't offer the expandability of say Wordpress when it comes to a CMS. But doesn't really need to...