A published store item means this item will be displayed on your store. Guests will be able to access that product from your website. If your item is "published" you'll see a green button.
A unpublished store item means it will not display on your store. However, if someone has the permalink they are still able to view the item online. If your item is "unpublished" you'll see a grey button.
If you're unpublishing a store item because you no longer carry it and do not want to delete the item from BentoBox, you should mark the item inventory as 0 to ensure no one can purchase it.
How to Publish & Unpublish Store Items
Click into your specific item.
Go to the Gear at the top left of the product. You'll see "Visibility" where you can mark it as either published or unpublished.